Iwo Bialynicki-Birula

Center for Theoretical Physics

In this paper we extend the Zeldovich formula, which was originally derived for the free electromagnetic field and was interpreted as the number of photons. We show that our extended formula gives a universal dimensionless measure of the overall strength of electromagnetic fields: free fields and fields produced by various sources, in both the clas...

The picture is showing the simplified intuitive approximate stability diagram of the motion of the particle with the Hamiltonian H = p_x^2/2 + p_y^2/2 + a omega^2 x^2/2 + b omega^2 y^2/2 - omega (x p_y - y p_x) which can be deduced without any calculations from the behavior of the mechanical models of the Trojan Wave Packet confinement mechanism in...

We derive new solutions of the Schr"odinger, Klein-Gordon and Dirac equations which describe the motion of particles in a uniform magnetic field. In contrast to the well known stationary solutions, our solutions exhibit the behavior of quantum particles which very closely resembles classical helical trajectories. These solutions also serve as an il...

We derive new solutions of the Schr\"odinger equation which describe the motion of particles in the Penning trap. These solutions are direct counterparts of classical orbits. They are obtained by injection of classical trajectories into the wave functions of stationary solutions.

We derive new solutions of the Schr\"odinger, Klein-Gordon and Dirac equations which describe the motion of particles in a uniform magnetic field. In contrast to the well known stationary solutions, our solutions exhibit the behavior of quantum particles which very closely resembles classical helical trajectories. These solutions also serve as an i...

In this work we extend the Zeldovich formula \cite{zeld}, which was originally derived for the free electromagnetic field and was interpreted as the number of photons. We show that our extended formula gives a universal dimensionless measure of the overall strength of electromagnetic fields: free fields and fields produced by various sources, in cl...

We present further arguments which show that the backflow has a universal character. It is not restricted to quantum theory and it appears in many theories (quantum or classical). It is a general property of waves propagating in any number of dimensions.

Ehrenfest theorem is proven in relativistic quantum theory of charged particles, moving under the influence of an external electromagnetic field. In order to extend the classic Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton's second law. Our approach is charac...

We show that, contrary to the statements made by many authors, the backflow is not a nonclassical effect. The backflow is a characteristic feature of solutions of the wave equations: quantum and classical. We present simple solutions of the Dirac equation, Maxwell equations and equations of linearized gravity where the backflow phenomenon is clearl...

We show that, contrary to the statements made by many authors, the backflow is not a nonclassical effect. The backflow is a characteristic feature of solutions of the wave equations: quantum and classical. We present simple solutions of the Dirac equation, Maxwell equations and equations of linearized gravity where the backflow phenomenon is clearl...

It is shown that the claim that the velocity of the electron never exceeds the speed of light is invalid. The velocity of the energy flow, as defined by the author, becomes even infinite at some points. We also show that the proof of the nonexistence of the lower limit on the size of the electron wave function can be obtained from simple dimensiona...

Ehrenfest theorem is proven in relativistic quantum theory of charged particles, moving under the influence of an external electromagnetic field. In order to extend the classic Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton's second law. Our approach is charac...

It is pointed out that the authors of the paper Li et al (2020 New J. Phys. 22 113019) have misinterpreted the Dirac equation by not taking into account that it describes not only electrons but also positrons.

We show that in the greatly simplified model of mutually interacting electron-positron pairs and an electric field, time-crystal structures can spontaneously form. For a special choice of parameters, we find periodic fluctuations of the pair number and the electric field.

This work completes the program started by I. Bialynicki-Birula and Z. Bialynicka-Birula [Uncertainty relation for photons, Phys. Rev. Lett. 108, 140401 (2012); Heisenberg uncertainty relation for photons, Phys. Rev. A 86, 022118 (2012); Heisenberg uncertainty relation for relativistic electrons, New J. Phys. 21, 073036 (2019)] to derive the Heisen...

It is shown that the interpretation of the electron wave function as a classical field is untenable because the so called energy-density defined in \cite{seb} takes on negative values in some regions. The claim that the velocity of the electron never exceeds the speed of light is also invalid. The velocity, as defined by the author, becomes even in...

It is shown that in the greatly simplified model of the mutually interacting electric field and electron-positron pairs the time-crystal structures can spontaneously form. For a wide range of parameters we find a periodic modulation of the electric field and the pair number.

It is pointed out that the solutions of the Klein-Gordon and the Dirac equation derived in the paper addressed in this Comment (and many more solutions) may be obtained from generating functions.

New solutions of relativistic wave equations are obtained in a unified manner from generating functions of spinorial variables. The choice of generating functions as Gaussians leads to representations in the form of generalized fractional Fourier transforms. Wave functions satisfying the Dirac, Maxwell, and Weyl equations are constructed by simple...

This work completes the program started in \cite{bb1,bb2,bb3} to derive the Heisenberg uncertainty relation for relativistic particles. Sharp uncertainty relations for massive relativistic particles with spin 0 and spin 1 are derived. The main conclusion is that the uncertainty relations for relativistic bosons are markedly different from those for...

New solutions of relativistic wave equations are obtained in a unified manner from generating functions of spinorial variables. The choice of generating functions as Gaussians leads to representations in the form of generalized fractional Fourier transforms. Wave functions satisfying the Dirac, Maxwell, and Weyl equations are constructed by simple...

We show that the standard method of introducing the quantum description of the electromagnetic field—by canonical field quantization—is not the only one. We have chosen here the relativistic quantum mechanics of the photon as the starting point. The treatment of photons as elementary particles merges smoothly with the description in terms of the qu...

We show that the standard method of introducing the quantum description of the electromagnetic field -- by canonical field quantization -- is not the only one. We have chosen here the relativistic quantum mechanics of the photon as the starting point. The treatment of photons as elementary particles merges smoothly with the description in terms of...

Berry phase is a very general concept. It is applied here to families of solutions of the Dirac equation with different values of spin. The value of the Berry phase in the spin space is given by the same expression as was found before in the momentum space.

In our Comment we question the validity of the claim made by Campos and Cabrera [Phys. Rev. Res. 2, 013051 (2020)] that their solutions of the Dirac equation in an external time-dependent electromagnetic field describe beams of electrons. In every time-dependent field, no matter how weak, which has an infinite time duration, there is continuous ele...

In our Comment we question the validity of the claim made by the authors of \cite{cc} that their solutions of the Dirac equation in an external {\em time-dependent} electromagnetic field describe beams of electrons. In every time-dependent field, no matter how weak, which has {\em infinite} time duration, there is a continuous electron-positron pai...

Analytical solutions of the Dirac equation in an external electromagnetic field are found such that according to the field-theoretic interpretation electron-positron pairs are trapped for a period of time. The naive one-particle interpretation of the Dirac wave function fails in this case completely. Simple electromagnetic field which produces this...

The purpose of this article is to show that the standard method of introducing the quantum description of the electromagnetic field -- by field quantization -- is not the only one. We choose instead relativistic quantum mechanics of photons as a starting point. Our present understanding of the nature of photons significantly differs from what has b...

The authors of this paper [M. Pandit et al., Phys. Rev. A 100, 012131 (2019)] proposed an alternative formulation of the uncertainty relation in quantum mechanics in terms of the Lipschitz constants. They evaluated these constants in various cases and decided that the product of the square roots of Lipschitz constants for position and momentum prob...

All known solutions of the Dirac equation describing states of electrons endowed with angular momentum are very far from our notion of the electron as a spinning charged bullet because they are not localized in the direction of propagation. We present here normalizable analytic exact solutions, eigenstates of the total angular momentum component Mz...

It is shown that the helicity amplitudes can be used to describe and analyze the properties of the electromagnetic field in classical and in quantum theory. On the one hand they embody the relativistic content of electromagnetic theory. On the other hand they give a concise description of such experimentally important notions as polarization, the S...

The Heisenberg uncertainty relation is derived for relativistic electrons described by the Dirac equation. The standard nonrelativistic lower bound 3/2ℏ is attained only in the limit and the wave function that reproduces this value is singular. At the other end, in the ultrarelativistic limit, the bound is the same as that found before for photons.

DOI:https://doi.org/10.1103/PhysRevLett.122.159301

The Heisenberg uncertainty relation is derived for relativistic electrons described by the Dirac equation. The standard nonrelativistic lower bound $3/2\hbar$ is attained only in the limit and the wave function that reproduces this value is singular. At the other end, in the ultrarelativistic limit, the bound is the same as that found before for ph...

DOI:https://doi.org/10.1103/PhysRevLett.122.089301

All known solutions of the Dirac equation describing states of electrons endowed with angular momentum are very far from our notion of the electron as a spinning charged bullet because they are not localized in the direction of propagation. We present here analytic exact solutions, eigenstates of the total angular momentum component $M_z$, that com...

Trapping of bodies by waves is extended from electromagnetism to gravity. It is shown that gravitational waves endowed with angular momentum may accumulate near its axis all kinds of cosmic debris. The trapping mechanism in both cases can be traced to the Coriolis force associated with the local rotation of the space metric. The same mechanism caus...

Trapping of bodies by waves is extended from electromagnetism to gravity. It is shown that gravitational waves endowed with angular momentum may accumulate near its axis all kinds of cosmic debris. The trapping mechanism in both cases can be traced to the Coriolis force associated with the local rotation of the space metric. The same mechanism caus...

It is shown that the description of light beams in terms of the corresponding photon quantum numbers elucidates the properties of these beams. In particular, this description shows that the helicity quantum number plays the fundamental role. This mode of description is applied to twisted and knotted electromagnetic waves. We concentrate on the case...

It is shown that photon helicity quantum number plays the fundamental role in the description of both twisted and knotted electromagnetic waves. We concentrated on the cases where photon wave functions are eigenfunctions of one component of angular momentum. The role of the photon wave function in momentum representation is emphasized because its k...

Quantum mechanics of photons is derived from the theory of representations of the Poincar\'e group developed by Wigner. This theory places helicity as the most fundamental property; angular momentum and polarization are secondary characteristics. The properties of the beams of light are shown to be fully determined by the quantum states of the phot...

DOI:https://doi.org/10.1103/PhysRevLett.119.029501

The propagation of electromagnetic waves trapped within dielectric and magnetic layers is considered. The description within the three-dimensional theory is compared with the simplified analysis in two dimensions. Two distinct media configurations with different topology are dealt with: a plane slab and a hollow cylinder. Choosing the appropriate v...

There are important differences between the nonrelativistic and relativistic description of electron beams. The orbital angular momentum quantum number cannot be used to specify the wave functions in the relativistic case. In this Letter we introduce analytic solutions of the Dirac equation in the form of exponential wave packets and we argue that...

The motion of a neutral atom endowed with a magnetic moment interacting with the magnetic field is determined from the Ehrenfest-like equations of motion. These equations for the average values of the translational and spin degrees of freedom are derived from the Schr\"odinger-Pauli wave equation and they form a set of nine coupled nonlinear evolut...

Spinorial formalism is used to map every electromagnetic wave into the gravitational wave (within the linearized gravity). In this way we can obtain the gravitational counterparts of Bessel, Laguerre-Gauss, and other light beams carrying orbital angular momentum.

The fluctuations of spacetime geometries at finite temperature are evaluated within the linearized theory of gravity. These fluctuations are described by the probability distribution of various configurations of the gravitational field. The field configurations are described by the linearized Riemann-Weyl tensor. The probability distribution has a...

The changes in the cosmic microwave background (CMB) spectrum seen as an increase of temperature due to a strong magnetic field are determined and their influence on the polarization of the radiation is exhibited. The effect is due to the coupling of the CMB photons to the magnetic field in the QED vacuum via the interaction with virtual pairs. In...

It is pointed out that there exists an unambiguous definition of locality that enables one to distinguish local and nonlocal quantities. Observables of both types coexist in quantum optics but one must be very careful when attempting to measure them. A nonlocal observable which formally depends on the spatial position $\bi r$ cannot be {\em locally...

Original definition of the Wigner function can be extended in a natural manner to relativistic domain in the framework of quantum field theory. Three such generalizations are described. They cover the cases of the Dirac particles, the photon, and the full electromagnetic field.

It is shown that Nambu dynamics can be generalized to any number of dimensions by replacing the 0(3) algebra, a prominent feature of Nambu's formulation, by an arbitrary Lie algebra. For the infinite dimensional algebra of rotations in phase space one obtains quantum mechanics in the Weyl-Wigner representation from the generalized Nambu dynamics. A...

Uncertainty relations for light pulses found in [Phys. Rev. A {\bf 86}, 022118 (2012)] were derived in the three-dimensional case which emphasized the localization in a volume. Here we derive the uncertainty relation for light beams in the two-dimensional plane perpendicular to the direction of the beam propagation which is more interesting for rea...

We construct analytically, a new family of null solutions to Maxwell's equations in free space whose field lines encode all torus knots and links. The evolution of these null fields, analogous to a compressible flow along the Poynting vector that is shear free, preserves the topology of the knots and links. Our approach combines the construction of...

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General solutions of the Klein-Gordon equation satisfying the Dirichlet boundary conditions on one fixed wall and the other wall moving with a constant velocity are derived. These solutions are specified by an arbitrary periodic function whose period is equal to twice the value of the rapidity of the moving wall. Choosing this function as a combina...

It is shown that the use of the Riemann-Silberstein (RS) vector greatly simplifies the description of the electromagnetic field both in the classical domain and in the quantum domain. In this review we describe many specific examples where this vector enables one to significantly shorten the derivations and make them more transparent. We also argue...

Recent developments in the angular momentum of light present fresh challenges to long established concepts and pave the way for new and wide-ranging applications. The scope for structured light such as optical vortices, in particular, now extends from microfluidics to quantum information. This is the first comprehensive edited collection dealing wi...

Recent developments in the angular momentum of light present fresh challenges to long established concepts and pave the way for new and wide-ranging applications. The scope for structured light such as optical vortices, in particular, now extends from microfluidics to quantum information. This is the first comprehensive edited collection dealing wi...

A Reply to the Comment by Z. Wang, C.-D. Xiong, and A. Qui.

The idea to base the uncertainty relation for photons on the electromagnetic energy distribution in space enabled us to derive a sharp inequality that expresses the uncertainty relation [Phys. Rev. Lett. {\bf 108}, 140401 (2012)]. An alternative version of the uncertainty relation derived in this paper is closer in spirit to the original Heisenberg...

The uncertainty relation for the photons in three dimensions that overcomes the difficulties caused by the nonexistence of the photon position operator is derived in quantum electrodynamics. The photon energy density plays the role of the probability density in configuration space. It is shown that the measure of the spatial extension based on the...

The term rotational frequency shift (RFS) has been used in different contexts and it was given different meanings. Other terms have also been used (azimuthal Doppler shift, angular Doppler shift) to describe various related phenomena. In this article we stick to the meaning of the rotational frequency shift given by us in Phys. Rev. Lett.78, 2539 (...

There exist two well established methods to trap charged particles: the Penning trap and the Paul trap. The subject of this article is to present a third mechanism for trapping charged particles - trapping by beams of electromagnetic radiation. The essential role is played by the electric field configuration in the plane perpendicular to the beam a...

Uncertainty relations have become the trademark of quantum theory since they were formulated by Bohr and Heisenberg. This review covers various generalizations and extensions of the uncertainty relations in quantum theory that involve the Rényi and the Shannon entropies. The advantages of these entropic uncertainty relations are pointed out and the...

Exact analytical solutions are presented for the time evolution of the density of pairs produced in the QED vacuum by a uniform electric field that is adiabatically switched on starting at minus infinity. Pair production is described by the Dirac-Heisenberg-Wigner function introduced before [Phys. Rev. D 44, 1825 (1991)]. The explicit solution is o...

It is shown that the photon picture of the electromagnetic field enables one to determine unambiguously the splitting of the total angular momentum of the electromagnetic field into the orbital part and the spin part.

Exact analytical solutions are presented for the time evolution of the density of pairs produced in the QED vacuum by a time-independent, uniform electric field. The mathematical tool used here to describe the pair production is the Dirac-Heisenberg-Wigner function introduced before [Phys. Rev. D 44, 1825 (1991)]. The initial value problem for this...

We dedicate this work to the memory of Krzysztof Wódkiewicz. Krzysztof was not only a brilliant scientist but also a devoted teacher and we are certain that he would be happy to know that this paper is based on the diploma thesis of the first author. a b s t r a c t Photon production by an oscillating medium predicted in [I. Bialynicki-Birula, Z. B...

We prove that the inequality used recently by Wilk and W{\l}odarczyk [Phys. Rev. A {\bf 79}, 062108 (2009)] to find a better lower bound in the uncertainty relations for the R\'enyi entropies is invalid. Thus, the problem of improving the bound given in our paper [Phys. Rev. A {\bf 74}, 052101 (2006)] remains unsolved.

We propose a simple three-body model of an atom in which one electron on a circular Rydberg orbit is treated as an independent particle and the remaining core electrons are collectively described as a single object. Within this model we predict the existence of stable deformed states of atoms. The deformation is generated by a bootstrap mechanism....

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Photons cannot be localized in a sharply defined region. The expectation value of their energy density and the photon number density can only be approximately localized, leaving an exponential tail. We show that one may sharply localize either electric or magnetic (but not both) footprints of photons and only momentarily. In the course of time evol...

We show that oscillations of a homogeneous medium with constant material coefficients produce pairs of photons. Classical analysis of an oscillating medium reveals regions of parametric resonance where the electromagnetic waves are exponentially amplified. The quantum counterpart of parametric resonance is an exponentially growing number of photons...

Gravitating bodies in motion, regardless of their constitution, always produce electromagnetic radiation in the form of photon pairs. This phenomenon is an analog of the radiation caused by the motion of dielectric (or magnetic) bodies. It is a member of a wide class of phenomena named dynamical Casimir effects, and it may be viewed as the squeezin...

We present Heisenberg's equation of motion for the radial variable of a free non-relativistic particle in D dimensions. The resulting radial force consists of three contributions: (i) the quantum fictitious force which is either attractive or repulsive depending on the number of dimensions, (ii) a singular quantum force located at the origin, and (...

quantum field theory;gravitation;canonical quantization;twistors and relativistic wave equations

Systematic description of a spin one-half system endowed with magnetic moment or any other two-level system (qubit) interacting with the quantized electromagnetic field is developed. This description exploits a close analogy between a two-level system and the Dirac electron that comes to light when the two-level system is described within the forma...

Quantum mechanical uncertainty relations for the position and the momentum and for the angle and the angular momentum are expressed in the form of inequalities involving the Rényi entropies. These uncertainty relations hold not only for pure but also for mixed states. Analogous uncertainty relations are valid also for a pair of complementary observ...

Properties of the Schrödinger equation with the logarithmic nonlinearity are briefly described. This equation possesses soliton-like solutions in any number of dimensions, called gaussons for their Gaussian shape. Excited, stationary states of gaussons of various symmetries, in two and three dimensions are found numerically. The motion of gaussons...

We compare and contrast five measures of phase uncertainty of a quantum state corresponding to a single mode of the electromagnetic field. The basis of this study are the states which minimize a particular measure for a fixed number of Fock states and normalization. We find these optimal states and study their characteristic properties. These optim...

New uncertainty relations based on the information entropy are reviewed and contrasted with the traditional uncertainty relations, which were based on the dispersions of the the physical variables. Improved lower bounds are given for the position-momentum and the angle-angular momentum pairs. Novel uncertainty relation for the angular distribution...

Quantum-mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the Rényi entropies. The proof of these inequalities requires the use of the exact expression for the (p,q)-norm of the Fourier transformation derived by Babenko and Beckner. Analogous uncertainty relations are derived for angle an...

We show that in addition to well known Bessel, Hermite-Gauss, and Laguerre-Gauss beams of electromagnetic radiation, one may also construct exponential beams. These beams are characterized by a fall-off in the transverse direction described by an exponential function of rho. Exponential beams, like Bessel beams, carry definite angular momentum and...

Electromagnetic beams of radiation endowed with orbital angular momentum have embedded vortex lines. These electromagnetic vortices act as beam guides for charged particles. Exact solutions of the classical (Lorentz) and quantum (Schroedinger and Dirac) equations, derived in Phys. Rev. Lett. 93, 20402 (2004), exhibit such a behavior. In the present...

Our present understanding of the nature of photons significantly differs from what has been known years ago when the concept of a photon has only been emerging. Unfortunately, very little of this knowledge trickles to those students who do not specialize in theoretical physics. In this lecture, in addition to giving a historical perspective on the...

Electromagnetic waves with phase defects in the form of vortex lines combined with a constant magnetic field are shown to pin down cyclotron orbits (Landau orbits in the quantum mechanical setting) of charged particles at the location of the vortex. This effect manifests itself in classical theory as a trapping of trajectories and in quantum theory...

All beams of electromagnetic radiation are made of photons. Therefore, it is important to find a precise relationship between the classical properties of the beam and the quantum characteristics of the photons that make a particular beam. It is shown that this relationship is best expressed in terms of the Riemann–Silberstein vector – a complex com...

Photon wave function is a controversial concept. Controversies stem from the fact that photon wave functions can not have all the properties of the Schroedinger wave functions of nonrelativistic wave mechanics. Insistence on those properties that, owing to peculiarities of photon dynamics, cannot be rendered, led some physicists to the extreme opin...

The problem of the linearity of the Schrödinger equation is described from a historical perspective. It is argued that the Schrödinger picture on which this equation is based cannot be retained in relativistic quantum theory. A closer analysis of realistic experiments might offer a clue how to modify the evolution equation for the state vectors in...

Previous work [I. Bialynicki-Birula, Phys. Rev. Lett. {\bf 93}, 20402 (2004)] is extended to cover more realistic examples of electromagnetic waves, viz. the Bessel beams. It is shown that electrons may be guided by a Bessel beam with nonvanishing orbital angular momentum. The mechanism for trapping the electrons near the electromagnetic vortex lin...

Modeling Reality covers a wide range of fascinating subjects, accessible to anyone who wants to learn about the use of computer modeling to solve a diverse range of problems, but who does not possess a specialized training in mathematics or computer science. The material presented is pitched at the level of high-school graduates, even though it cov...

We study the influence of the nonlinearity in the Schrödinger equation on the motion of quantum particles in a harmonic trap. In order to obtain exact analytic solutions, we have chosen the logarithmic nonlinearity. The unexpected result of our study is the existence in the presence of nonlinearity of two or even three coexisting Gaussian solutions...

It is shown that in an anisotropic harmonic trap that rotates with the properly chosen rotation rate, the force of gravity leads to a resonant behavior. Full analysis of the dynamics in an anisotropic, rotating trap in 3D is presented and several regions of stability are identified. On resonance, the oscillation amplitude of a single particle, or o...