Imaginary-time method in quantum mechanics and field theory

An imaginary-time method was developed for calculating the probability of particle transmission through smooth barriers variable with time. Within the imaginary-time method, the tunneling process is described by using classical equations of motion written in terms of an imaginary time (t → it), while the probability of tunneling is determined by the imaginary part of the action functional, this imaginary part being calculated along the subbarrier particle trajectory. The fundamentals of the imaginary-time method are surveyed, and its applications in the theory of atomic-state ionization under the effect of constant electric and magnetic fields that have various configurations, in the field of intense monochromatic laser radiation and of an ultrashort electromagnetic pulse, in the process of Lorentz ionization of atoms and ions during their motion in a strong magnetic field, etc., are outlined. The applications of the imaginary-time method in relativistic cases—for example, in the theory of ionization of levels of multiply charged ions whose binding energy is commensurate with the electron rest energy—and in quantum field theory (Schwinger effect, which consists in the production of electron-positron pairs from a vacuum by a superstrong external field) are briefly described. Particular attention is given to methodological issues and details of the imaginary-time method that are of importance in solving specific physics problems, but which are usually skipped in original publications.     

The imaginary-time method for relativistic problems

A relativistic version of the imaginary-time method is presented. The method is used to calculate the probability w of ionization of a bound state by electric and magnetic fields of various configurations (including the case when the binding energy E _b is comparable to mc ₂). The formulas cover as limiting cases both the ionization of nonrelativistic bound systems (atoms and ions) and the case E _b =2mc ₂, when w equals the probability of electron-positron pair production from the vacuum in the presence of a strong field. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 4, 213–218 (25 August 1997)     

Deep impurity-center ionization by far-infrared radiation

An analysis is made of the ionization of deep impurity centers by high-intensity far-infrared and submillimeter-wavelength radiation, with photon energies tens of times lower than the impurity ionization energy. Within a broad range of intensities and wavelengths, terahertz electric fields of the exciting radiation act as a dc field. Under these conditions, deep-center ionization can be described as multiphonon-assisted tunneling, in which carrier emission is accompanied by defect tunneling in configuration space and electron tunneling in the electric field. The field dependence of the ionization probability permits one to determine the defect tunneling times and the character of the defect adiabatic potentials. The ionization probability deviates from the field dependence e(E) ∝ exp(E ²/E _c ² ) (where E is the wave field, and E _c is a characteristic field) corresponding to multiphonon-assisted tunneling ionization in relatively low fields, where the defects are ionized through the Poole-Frenkel effect, and in very strong fields, where the ionization is produced by direct tunneling without thermal activation. The effects resulting from the high radiation frequency are considered and it is shown that, at low temperatures, they become dominant. Fiz. Tverd. Tela (St. Petersburg) 39, 1905–1932 (November 1997)     

Ionization of atoms in electric and magnetic fields and the imaginary time method

A semiclassical theory is developed for the ionization of atoms and negative ions in constant, uniform electric and magnetic fields, including the Coulomb interaction between the electron and the atomic core during tunneling. The case of crossed fields (Lorentz ionization) is examined specially, as well as the limit of a strong magnetic field. Analytic equations are derived for arbitrary fields ℰ and ℋ that are weak compared to the characteristic intraatomic fields. The major results of this paper are obtained using the “imaginary time” method (ITM), in which tunneling is described using the classical equations of motion but with purely imaginary “time.” The possibility of generalizing the ITM to the relativistic case, as well as to states with nonzero angular momentum, is pointed out. Zh. éksp. Teor. Fiz. 113, 1579–1605 (May 1998)     

Hawking radiation of <Emphasis Type="Italic">E</Emphasis> < <Emphasis Type="Italic">m</Emphasis> massive particles in the tunneling formalism

We use the tunneling formalism to calculate the Hawking radiation of massive particles. For E ≥ m, we recover the traditional result, identical to the massless case. But E < m particles can also tunnel across the horizon in a Hawking process. We study the probability for detecting such E < m particles as a function of the distance from the horizon and the energy of the particle in the tunneling formalism. We derive a general formula and obtain simple approximations in the near-horizon limit and in the limit of large radii.     

Effect of a magnetic field on thermally activated tunneling ionization of impurity centers in semiconductors

An expression for the probability of thermally activated tunneling ionization in an electric field in the presence of a magnetic field is obtained. It is shown that the logarithm of the ionization probability is proportional to the squared electric field, and the coefficient of proportionality decreases with increasing magnetic field. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 10, 763–767 (25 November 1998)     

Effect of a magnetic field on the ionization of atoms

The ionization probability of an atomic s state under the action of static electric and magnetic fields is calculated taking into account the Coulomb interaction between the escaping electron and the atomic core. The structure of the perturbation series for the energy of the level is investigated and the asymptotic behavior of the higher orders of the perturbation theory is found. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 6, 398–402 (25 March 1996)     

Double atomic photoeffect in the relativistic domain: Angular and energy distributions of photoelectrons

We study the double ionization of the atomic K-shell by a single photon in the relativistic energy domain. The differential and total cross sections of the process are calculated. It is shown that the ratio of the cross sections of double and single ionization increases with the photon energy, tending to the limit 0.34/Z ², where Z is the atomic number or the nuclear charge. The formulas are found to be valid for Z≫1 and αZ≪1, where α=1/137 is the fine-structure constant. Zh. éksp. Teor. Fiz. 114, 1537–1554 (November 1998)     

Pure Bound Field theory and the Decay of Moun in Meso-Atoms

In the present paper we consider the electrically bound quantum particles within the framework of pure bound field theory (PBFT) (Kholmetskii, A.L., et al.: Phys. Scr. 82, 045301 (2010)), which explicitly takes into account the non-radiative nature of electromagnetic (EM) field generated by bound charges in the stationary energy states, and evokes the appropriate modifications of bound EM field, which secure the total momentum conservation law for the isolated system “electron plus nucleus” in the absence of EM radiation. Such a PBFT gives the same gross as well as fine structure of atomic energy levels, as those furnished by the common approach, but implies a scaling transformation of radial coordinates. In this paper we find out that in the classical limit this transformation reflects the dependence of time rate for the orbiting electron on the electric potential of the binding EM field in addition to relativistic dependence on its Lorentz factor. We show that this effect completely eliminates the available up to date discrepancy between calculated and experimental data on the decay rate of bound muon in meso-atoms. We emphasize that the revealed dependence of time rate of quantum electrically bound particles on the electric potential represents the specific effect of PBFT, and, in general, is not extended to the classical world.     

Contribution to the theory of Lorentzian ionization

The probability w _L of Lorentzian ionization, which arises when an atom or ion moves in a constant magnetic field, is calculated in the quasiclassical approximation. The nonrelativistic (v≲e ²/ℏ=1, v is the velocity of the atom) and ultrarelativistic (v→c=137) cases are examined and the stabilization factor S, which takes account of the effect of the magnetic field on tunneling of an electron, is found. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 5, 391–396 (10 March 1997)     

Beam-plasma discharge in the propagation of a long-pulse relativistic electron beam in a medium-pressure rarefied gas

A theoretical model is given, along with a numerical analysis of the evolution of beam-plasma discharge in the propagation of a long-pulse relativistic electron beam in a rarefied gas at medium pressure. It is shown that the self-stabilization of beam-plasma discharge as a result of longitudinal inhomogeneity of the density of the discharge plasma makes it possible for the beam to traverse the beam chamber with relatively low total energy losses, including ionization losses and energy losses in the generation of oscillations. During the dissociative recombination of electrons and ions of the discharge-driven plasma, heat is released and spent in raising the temperature of the gas. The investigated collective-discharge mechanism underlying heating of the gas for a relativistic beam can be more efficient than the classical heating mechanism due to ionization losses of the beam in pair collisions of its electrons with gas particles. Zh. Tekh. Fiz. 67, 94–98 (May 1997)     

Multiphoton atomic ionization in the field of a very short laser pulse

Closed analytic expressions are derived for the probability of multiphoton atomic and ionic ionization in a variable electric field ?(t), which are applicable for arbitrary Keldysh parameters γ. Dependencies of the ionization probability and photoelectron pulse spectrum on the shape of a very short laser pulse are analyzed. Examples of pulse fields of various forms, including a modulated light pulse with a Gaussian or Lorentz envelope, are considered in detail. The interference effect in the photoelectron energy spectrum during atomic ionization by a periodic field of a general form is examined. The range of applicability of the adiabatic approximation in the multiphoton ionization theory is discussed. The imaginary time method is used in the calculations, which allows the probability of particle tunneling through oscillating barriers to be effectively calculated.     

Generalizations of Quantum Mechanics Induced by Classical Statistical Field Theory

No Heading We show that the Dirac-von Neumann formalism for quantum mechanics can be obtained as an approximation of classical statistical field theory. This approximation is based on the Taylor expansion (up to terms of the second order) of classical physical variables – maps f : Ω → R, where Ω is the infinite-dimensional Hilbert space. The space of classical statistical states consists of Gaussian measures ρ on Ω having zero mean value and dispersion σ²(ρ) ≈ h. This viewpoint to the conventional quantum formalism gives the possibility to create generalized quantum formalisms based on expansions of classical physical variables in the Taylor series up to terms of nth order and considering statistical states ρ having dispersion σ²(ρ) = hⁿ (for n = 2 we obtain the conventional quantum formalism).     

On Three Magnetic Relativistic Schr?dinger Operators and Imaginary-Time Path Integrals

Three magnetic relativistic Schr?dinger operators are considered corresponding to the classical relativistic Hamiltonian symbol with magnetic vector and electric scalar potentials. We discuss their difference in general and their coincidence in the case of constant magnetic fields, as well as whether they are covariant under gauge transformation. Then results are surveyed on path integral representations for their respective imaginary-time relativistic Schr?dinger equations, i.e. heat equations, by means of the probability path space measure coming from the Lévy process concerned.     

Resonance split of ballistic conductance peaks in electric and magnetic superlattices

Resonant peak splitting for ballistic conductance in finite electric superlattices (ES) and magnetic superlattices (MS) was investigated theoretically. It is shown that, for electron tunneling through the ES (MS) of identical n electric (magnetic) barriers, the resonance split of the conductance peak is (n–1)-fold; while for electron tunneling through the ES (MS) made of two different barriers, one resonant window of the former splits into two subwindows, within each of which the resonance split is (m–1)-fold, where m is the number of the renormalized building blocks consisting of two different barriers of the latter. Received 15 February 2000     

The Poole–Frenkel effect in a dielectric under nanosecond irradiation by an electron beam with moderate or high current density

Processes of electron trapping and detrapping determine in many respects intense processes arising in dielectric and delayed by 1–100 ns from the irradiation pulse of a high-power electron beam, such as electron emission, electric discharge in the bulk of the dielectric, flashover, and electric breakdown. A model of charged donor center ionization in a dielectric exposed to a strong electric field is constructed. The model takes into account 1) the energy spectrum of the charged donor center in the dielectric, 2) the semiclassical state density in the donor center, 3) spontaneous emission of phonons by the electron localized in the donor center, 4) increase in the kinetic energy of the electron (heating) in the external electric field, 5) electron tunneling through a potential barrier and its reflection from the barrier depending on the external field intensity, and 6) thermal fluctuations of energy of the electron localized in the donor center. The probability of charged donor center ionization in the dielectric per unit time is calculated. In weak fields, the field dependence of the ionization probability almost coincides with that for the Poole–Frenkel theory. In strong fields, the contribution of electron heating to the external electric field is the deciding factor. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 10–16, December, 2008.     

An analytical solution of the Klein–Fock–Gordon equation for a 2D pion atom moving in a constant uniform magnetic field

The Klein–Fock–Gordon equation is solved for a 2D pion moving in a constant uniform magnetic field. A relativistic energy spectrum is calculated for fixed values of the angular momentum and magnetic field Н. An analysis of the results of these calculations allows us to conclude that the Klein–Fock–Gordon equation, unlike the Schr?dinger equation, cannot describe the energy of the particle s-state in the magnetic field. It is elucidated that a correction for the relativistic energy level caused by the constant magnetic field is noticeable for the magnetic field H > 100. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 91–96, March, 2009.     

Theory of the oscillatory dependence of the conductivity of two-component dielectric composites at room temperatures

It is predicted that at room temperatures a hopping mechanism of charge transfer plays a very important role and leads to temperature oscillations of the conductivity σ(T) of a dielectric composite. The dependence of the conductivity σ(ω) on the frequency of an alternating electric field is calculated. The relation obtained can be used to determine, first, the electron relaxation times and, second, and more importantly, the frequency of electron tunneling through the dielectric matrix from measurements of the conductivity in various frequency ranges. Zh. Tekh. Fiz. 69, 31–34 (March 1999)     

Investigations on the Fermions Tunneling Radiation from?the?Charged Rotating Kaluza-Klein Spacetime

The study of Hawking radiation of fermions via tunneling is a hot spot of current topics in black hole physics. By constructing a set of appropriate matrices γ ^μ for general covariant Dirac equation, the tunneling effect of Kaluza-Klein spacetime was deeply studied. For spacetimes with different dimensions, constructing a set of appropriate γ ^μ matrices for general covariant Dirac equation is an important technique for fermions tunneling method. As a result, the tunneling probability of Dirac particles and the expected Hawking temperature of the spacetime were successfully recovered.     

Tunneling method for Hawking radiation in the Nariai case

We revisit the tunneling picture for the Hawking effect in light of the charged Nariai manifold, because this general relativistic solution, which displays two horizons, provides the bonus to allow the knowledge of exact solutions of the field equations. We first perform a revisitation of the tunneling ansatz in the framework of particle creation in external fields à la Nikishov, which corroborates the interpretation of the semiclassical emission rate \({\varGamma }_{emission}\) as the conditional probability rate for the creation of a couple of particles from the vacuum. Then, particle creation associated with the Hawking effect on the Nariai manifold is calculated in two ways. On the one hand, we apply the Hamilton–Jacobi formalism for tunneling, in the case of a charged scalar field on the given background. On the other hand, the knowledge of the exact solutions for the Klein–Gordon equations on Nariai manifold, and their analytic properties on the extended manifold, allow us a direct computation of the flux of particles leaving the horizon, and, as a consequence, we obtain a further corroboration of the semiclassical tunneling picture from the side of S-matrix formalism.