Curriculum Vitae Dmitry Maximovitch Gitman August 31, 2010

Contents

1 Personal information 1

2 Education and academic titles 2

3 Academic Positions 2

4 Didactic Activities 3

5 Orientation of students and post-docs 3

6 Books published and in progress 5

7 Selected articles 6

8 Number of publications and international citations 10

9 Coordination of projects and research group 10

10 Summary of the main scientific results obtained 11 10.1 Quantum field theory with external backgrounds ...... 11 10.2 General theory of constrained systems and their quantization ...... 15 10.3 Exact solutions of the relativistic wave equations and theory of self-adjoint extensions . 18 10.4 Path integrals; group theory in relativistic quantum mechanics and field theory; semiclassical methods and coherent states ...... 22 10.5 Classical and pseudoclassical relativistic particle models and their quantization . . . . . 25 10.6 Theoryofhigherspins ...... 27 10.7 Theory of two and four levels systems and applications to the quantum computation . . 28 10.8 Quantum mechanics and field theory in non-commutative spaces ...... 29 10.9 QuantumStatistics...... 30 10.10Othersubjects ...... 31

1 Personal information

Born on July 2, 1944, Tashkent, Uzbekistan - USSR Brazilian Citizenship Languages: Russian, English, Portuguese and German Address: Instituto de Física da USP, Departamento de Física Nuclear, Instituto de Física, Universidade de São Paulo, C.P. 66318, CEP 05315-970, São Paulo, SP, Brasil Telephone: 55-11-3091-6948. E-mail: [email protected], [email protected] Homepage: http://www.dfn.if.usp.br/pesq/tqr/pt/gitman

1 2 Education and academic titles

1961-1966: Graduation and Master’s degree — Department of Physics of Tomsk State University • — Russia. Received a medal and the “red diploma”. Master’s Thesis: “Integral Equations for Distribution Functions in Statistical Mechanics”.

1966-1969: Doctorate, Tomsk State University — Russia. 1969: Ph.D. Title — Candidate of Physi- • cal and Mathematical Sciences, obtained with the thesis “Variational Principles in Quantum Statistics” — Tomsk State University, Russia. Diploma MFM No. 011435, State Higher Attesta- tion Commission, Moscow, January 16, 1970. Received oﬃcial favorable opinions from Professors V.L. Pokrovsky - Landau Institute, Chernogolovka: M. Gaigasian — Tomsk State University and Prof. A.A. Sokolov, Moscow State University. This title was recognized by University of São Paulo in June 5, 1996 as equivalent to Doctor accordingly to document number 94.1.997.43.4.

1975: Received the academic degree of Docent of the Chair of Theoretical and Mathe- • matical Physics. Diploma MDC no. 096071, “Higher Attestation Commission”, Moscow, April 1, 1976.

1979: Received the title “Doctor in Sciences”, (the highest scientiﬁc degree in Physics and • Mathematics given in Russia) with the thesis “Problems of External Fields in Quantum Electrodynamics”, Institute of Nuclear Physics of Novosibirsk. Diploma FM No. 001066, “Higher Attestation Commission”, Moscow, April 18, 1980. Received oﬃcial favorable opinions from Professors F.A. Berezin, Moscow State University, B.N. Baier, Institute of Nuclear Physics, Novosibirsk, V.N. Barbashov, International Institute of Nuclear Research, Dubna, and V.I. Manko, Lebedev Physical Institute, Moscow. This title was recognized by University of São Paulo in June 5, 1996 as equivalent to Full Professor in accordance with document No. 94.1.1005.43.5

1981: Received the academic degree “Cathedratic Professor of the chair of Theoretical • Physics”. Approved by the Russian Ministry of Higher Education. Diploma PR No. 007429, Higher Attestation Commission, Moscow, June 26, 1981.

3 Academic Positions

1969-1970: Assistant Professor at the Department of Physics — Tomsk Institute of Automation • Control Systems and Radio Engineering (TIASUR), Tomsk, Russia.

1970-1975: Associate Professor at the Department of Physics — Tomsk Institute of Automation • Control Systems and Radio Engineering, Tomsk, Russia.

1975-1985: Cathedratic Professor of the chair of Mathematical Analysis — Tomsk State Pedagog- • ical University (TSPU), Tomsk, Russia.

1985-1992: Full Professor of the Department of Mathematics — Moscow Institute of Radio Engi- • neering, Electronics and Automation (MIREA), Moscow, Russia.

1992-1996: Professor, (RDIDP, level MS-6) at the Department of Mathematical Physics, Physics • Institute — University of São Paulo, Brazil.

1996-1998: Associate Professor (RDIDP, level MS-5) at the Department of Mathematical Physics, • Physics Institute — University of São Paulo, Brazil.

1998-: Full Professor (RDIDP, level MS-6) at the Department of Nuclear Physics, Physics Institute • — University of São Paulo, Brazil.

1995-: CNPq Research Productivity Scholarship level PQ-1A. •

2 4 Didactic Activities

Since 1969, gave undergraduate and graduate courses in many universities1.

In Russia 1. General Physics, 1969-1975, TIASUR

2. Quantum Mechanics, 1970-1975, TIASUR

3. Quantum Theory of the Solid State, 1972-1975, TIASUR

4. Electrodynamics, 1970-1975, TIASUR

5. Mathematical Physics, 1976-1985, TSPU

6. Mathematical Calculus, 1976-1982, TSPU, MIREA

7. Group Theory, 1976-1985, TSPU

8. Quantum Field Theory, 1976-1985, TSU

9. Quantization of Constrained Systems, 1980-1984, TSU

10. General Relativity, 1970-1975, TSU

In Brazil 1. Constrained systems theory, 1992, 1994, 2000, 2009, USP, for graduated students.

2. Path integrals in quantum mechanics and quantum ﬁeld theory, 1993, 1994, 1997, 1999, USP, for graduated students.

3. General relativity, 1993, 1996, 2007, 2009, USP, for graduated students.

4. Introduction to general relativity, 1995, 2000, 2008, USP, for undergraduate students.

5. Quantum Mechanics II, 1995, USP, for undergraduate students.

6. General Physics IV, 1996, 2005, USP, for undergraduate students.

7. General Physics III, 1998, 2001, 2002, 2003, 2006, 2007, USP, for undergraduate students.

8. General Physics I, 2004, 2005, USP, for undergraduate students.

5 Orientation of students and post-docs

Oriented the following students:

Masters (only information in Brazil) 1. João Luis Meloni Assirati. Dissertation: Covariant generalization of Weyl ordering and particle quantization, (University of São Paulo, São Paulo, September 2001)

2. Mario Cesar Baldiotti, Dissertation: One electron quantum states in a uniform magnetic ﬁeld, (University of São Paulo, São Paulo, May 2002)

3. Rodrigo Fresneda, Dissertation: Relativistic spinorial particle quantization in 2 + 1 dimensions, (University of São Paulo, São Paulo, August 2003)

1 TIASUR-Tomsk Institute of Automation Control Systems and Radio Engineering; TSPU-Tomsk State Pedagogical Universtity; MIREA-Moscow Institute of Radio Engineering, Electronics and Automation; TSU-Tomsk State University; USP-University of Sao Paulo 3 Doctorates 1. P.V. Bozrikov. Thesis: Motion of an electron in the quantized electromagnetic plane wave, (Tomsk State University, Tomsk, 1973);

2. P.M. Lavrov. Thesis: Processes with an electron in the quantized electromagnetic plane wave, (Tomsk State University, Tomsk, 1975);

3. Sh.M. Shvartsman. Thesis: Some quantum processes in intensive electromagnetic ﬁelds, (Tomsk State University, Tomsk, 1975);

4. V.M. Shachmatov. Thesis: Quantum processes with a relativistic charged particle, interacting with strong electromagnetic ﬁeld, (Azerbaijan State University, Baku, 1978);

5. S.P. Gavrilov. Thesis : Some problems of QED with an external ﬁeld, creating pairs, (Moscow State University, Moscow, 1981);

6. I.M. Lichtzier. Thesis: Green’s functions and one-loop eﬀective action in external gauge and gravitational ﬁelds, (Tomsk State University, Tomsk, 1988);

7. M.D. Noskov. Thesis: Problems of QED with intensive external ﬁelds, (Tomsk State University, Tomsk, 1989);

8. V.P. Barashev. Thesis: Problems of QED with unstable vacuum, (Tomsk State University, Tomsk, 1989);

9. A.L. Shelepin. Thesis: Group methods in quantum theory and coherent states, (Lebedev Physical Institute, Moscow, 1990).

10. Antonio Edson Gonçalves. Thesis: Pseudoclassical models and their quantizations, (University of São Paulo, 1995).

11. Wellington da Cruz. Thesis: Path integral representations of relativistic particle propagators, (University of São Paulo, 1995).

12. Paulo Barbosa Barros. Thesis: Some applications of Grassmannian path integrals in modern quantum theory, (University of São Paulo, São Paulo, 1998)

13. Jose Nemecio Acosta Jara. Thesis: Quantum theory of particle radiation in a solenoidal magnetic ﬁeld, (University of São Paulo, São Paulo, December 2002)

14. Andrei Smirnov, Thesis: The Dirac equation with a superposition of Aharonov-Bohm and colinear uniform magnetic ﬁelds, (University of São Paulo, São Paulo, august 2004)

15. Mario Cesar Baldiotti. Thesis: Analytic study and exact solutions of the spin equation, (University of São Paulo, São Paulo, June 2005)

16. Rodrigo Fresneda, Thesis: Some problems of quantization in theories with non-Abelian backgrounds and in non-commutative space-times, (University of São Paulo, São Paulo, October 2008)

17. Vladislav Kupriyanov, Thesis, Quantization of non-Lagrangian systems and non-commutative quantum mechanics, (University of São Paulo, São Paulo, march 2009)

18. João Luis Meloni Assirati, Thesis: Covariant quantization of mechanical systems. (University of São Paulo, São Paulo, april 2010)

4 Post-docs 1. 1997-1999: G. Fulop, Project: Reparametrization invariance and zero Hamiltonian phenomenon

2. 1997-1998: A.V. Galajinsky, Project: Classical and quantum dynamics of the theory of supersymmetric extended objects

3. 1997-1998: A. Deriglazov, Project: Problems of covariant formulation and quantization of constraint systems

4. 1998-2000: A. L. Shelepin, Project: Methods of harmonic analysis and of generalized regular representation for Poincare group in various dimensions

5. 2002-2009: Pavel Moshin, Project: Problems of Lagrangian and Hamiltonian BRST quantization of gauge theories

6. 2004: A. Smirnov, Project: The study of vacuum polarization in backgrounds with singularities

7. 2005-: Mário César Baldiotti, Project: Study of two and four level systems, in progress;

8. 2010-: Rodrigo Fresneda, Project: Quantization of theories in non-commutative space-times, in progress.

6 Books published and in progress

1. Exact Solutions of the Relativistic Wave Equations with V. G. Bagrov, I. M. Ternov et al. Nauka, Novosibirsk (1982) 144 pages (in russian)

2. Canonical Quantization of Fields with Constraints with I. V. Tyutin Nauka, Moscow (1986) 216 pages (in russian)

3. Exact Solutions of Relativistic wave Equations with V. G. Bagrov Kluwer Acad. Publish., Dordrecht, Boston, London (1990) 321 pages

5 4. Quantum Electrodynamics with Unstable Vacuum with E. S. Fradkin e Sh. M. Shvartsman Nauka, Moscow (1991) 294 pages (in russian)

5. Quantum Electrodynamics with Unstable Vacuum with E. S. Fradkin, Sh. M. Shvartsman Springer-Verlag, Berlin, Heidelberg, New-York, London, Paris, Hong-Kong, Barcelona, (1991) 300 pages

6. Quantization of Fields with Constraints with I. V. Tyutin Springer-Verlag, Berlin, Heidelberg, New-York, London, Paris, Hong-Kong, Barcelona, (1990) 292 pages

7. Physical Observables in Non-trivial Quantum Systems (Constructing self-adjoint operators and solving spectral problems), with I. V. Tyutin and B. Voronov, to be published in 2011 by Birkhäuser (Springer Verlag).

8. Relativistic Wave Equations with External Electromagnetic Fields and their Solu- tions, with V. G. Bagrov, to be published in 2012 by Kluwer Acad. Publish.

9. Classical Theory of Constrained Systems, with I. V. Tyutin, to be published in 2013 by Springer Verlag.

7 Selected articles

The following listed articles are a selection of his most important publications, together with the respective abstracts:

1. D.M. Gitman and A. Shelepin, Field on Poincaré Group and Quantum Description of Orientable Objects, Europ. Physical Journal C, 61, Issue1 (2009)111, DOI 10.1140/epjc/s10052-009-0954-x. We propose an approach to the quantum-mechanical description of relativistic orientable objects. It generalizes Wigner’s ideas concerning the treatment of nonrelativistic orientable objects (in particular, a nonrelativistic rotator) with the help of two reference frames (space-fixed and body- fixed). A technical realization of this generalization (for instance, in 3 + 1 dimensions) amounts to introducing wave functions that depend on elements of the Poincaré group G. A complete set of transformations that test the symmetries of an orientable object and of the embedding space belongs to the group Π = G G. All such transformations can be studied by considering a × generalized regular representation of G in the space of scalar functions on the group, f(x, z), that depend on the Minkowski space points x G/Spin(3, 1) as well as on the orientation variables ∈ given by the elements z of a matrix Z Spin(3, 1). In particular, the field f(x, z) is a generating ∈ 6 function of usual spin-tensor multicomponent fields. In the theory under consideration, there are four different types of spinors, and an orientable object is characterized by ten quantum numbers. We study the corresponding relativistic wave equations and their symmetry properties.

2. S.P. Gavrilov and D.M. Gitman, Consistency Restrictions on Maximal Electric-Field Strength in Quantum Field Theory, Phys. Rev. Lett. 101, 130403(4) (2008). QFT with an external background can be considered as a consistent model only if backreaction is relatively small with respect to the background. To find the corresponding consistency restrictions on an external electric field and its duration in QED and QCD, we analyze the mean energy density of quantized fields for an arbitrary constant electric field E, acting during a large but finite time T . Using the corresponding asymptotics with respect to the dimensionless parameter eET 2, one can see that the leading contributions to the energy are due to the creation of particles by the electric field. Assuming that these contributions are small in comparison with the energy density of the electric background, we establish the above-mentioned restrictions.

3. B.L. Voronov, D.M. Gitman, and I.V. Tyutin, The Dirac Hamiltonian with a superstrong Coulomb field, Theoretical and Mathematical Physics, 150(1) (2007) 34-72 (Translated from Teoretich- eskaya i Matematicheskaya Fizika, 150, No. 1, pp.41-84, 2007). We consider the quantum-mechanical problem of a relativistic Dirac particle moving in the Coulomb field of a point charge Ze. In the literature, it is often declared that a quantum- mechanical description of such a system does not exist for charge values exceeding the so-called 1 critical charge with Z = α− = 137 based on the fact that the standard expression for the lower bound state energy yields complex values at overcritical charges. We show that from the mathematical standpoint, there is no problem in defining a self-adjoint Hamiltonian for any value of charge. What is more, the transition through the critical charge does not lead to any quali- tative changes in the mathematical description of the system. A specific feature of overcritical charges is a non uniqueness of the self-adjoint Hamiltonian, but this non uniqueness is also characteristic for charge values less than the critical one (and larger than the subcritical charge with 1 Z = (√3/2)α− = 118). We present the spectra and (generalized) eigenfunctions for all self- adjoint Hamiltonians. The methods used are the methods of the theory of self-adjoint extensions of symmetric operators and the Krein method of guiding functionals. The relation of the constructed one-particle quantum mechanics to the real physics of electrons in superstrong Coulomb fields where multiparticle effects may be of crucial importance is an open question.

4. S.P. Gavrilov, D.M. Gitman, One-loop energy-momentum tensor in QED with electric-like background, Phys. Rev. D78, 045017(35) (2008). We have obtained non-perturbative one-loop expressions for the mean energy-momentum tensor and current density of Dirac’s field on a constant electric-like background. One of the goals of this calculation is to give a consistent description of back-reaction in such a theory. Two cases of initial states are considered: the vacuum state and the thermal equilibrium state. First, we perform calculations for the vacuum initial state. In the obtained expressions, we separate the contributions due to particle creation and vacuum polarization. The latter contributions are related to the Heisenberg—Euler Lagrangian. Then, we study the case of the thermal initial state. Here, we separate the contributions due to particle creation, vacuum polarization, and the contributions due to the work of the external field on the particles at the initial state. All these contributions are studied in detail, in different regimes of weak and strong fields and low and high temperatures. The obtained results allow us to establish restrictions on the electric field and its duration under which QED with a strong constant electric field is consistent. Under such restrictions, one can neglect the back-reaction of particles created by the electric field. Some of the obtained results generalize the calculations of Heisenberg—Euler for energy density to the case of arbitrary strong electric fields.

5. D.M. Gitman, I.V. Tyutin, Symmetries and physical functions in general gauge theory, Int. J. Mod. Phys.A, 21, No.2 (2006) pp. 327-360. The aim of the present article is to describe the symmetry structure of a general gauge (singular) theory, and, in particular, to relate the structure of gauge transformations with the constraint

7 structure of a theory in the Hamiltonian formulation. We demonstrate that the symmetry structure of a theory action can be completely revealed by solving the so-called symmetry equation. We develop a corresponding constructive procedure of solving the symmetry equation with the help of a special orthogonal basis for the constraints. Thus, we succeed in describing all the gauge transformations of a given action. We find the gauge charge as a decomposition in the orthogonal constraint basis. Thus, we establish a relation between the constraint structure of a theory and the structure of its gauge transformations. In particular, we demonstrate that, in the general case, the gauge charge cannot be constructed with the help of some complete set of first-class constraints alone, because the charge decomposition also contains second-class constraints. The above-mentioned procedure of solving the symmetry equation allows us to describe the structure of an arbitrary symmetry for a general singular action. Finally, using the revealed structure of an arbitrary gauge symmetry, we give a rigorous proof of the equivalence of two definitions of physi- cality condition in gauge theories: one of them states that physical functions are gauge-invariant on the extremals, and the other requires that physical functions commute with FCC (the Dirac conjecture).

6. S.P. Gavrilov, D.M. Gitman, and J.L. Tomazelli, Density matrix of a quantum field in a particle- creating background, Nucl. Phys. B 795 [FS] (2008) 645-677. We examine the time evolution of a quantized field in external backgrounds that violate the stability of vacuum (particle-creating backgrounds). Our purpose is to study the exact form of the final quantum state (the density operator at the final instant of time) that has emerged from a given arbitrary initial state (from a given arbitrary density operator at the initial time instant) in the course of evolution. We find a generating functional that allows one to obtain density operators for an arbitrary initial state. Averaging over states of the subsystem of antiparticles (particles), we obtain explicit forms of reduced density operators for the subsystem of particles (antiparticles). Analyzing one-particle correlation functions, we establish a one-to-one correspondence between these functions and the reduced density operators. It is shown that in the general case a presence of bosons (e.g., gluons) in the initial state increases the creation rate of the same type of bosons. We discuss the question (and its relation to the initial stage of quark-gluon plasma formation) whether a thermal form of one-particle distribution can appear even if the final state of the complete system is not in thermal equilibrium. In this respect, we discuss some cases when pair-creation by an electric-like field can mimic the one-particle thermal distribution. We apply our technics to some QFT problems in slowly varying electric-like backgrounds: electric, SU(3) chromoelectric, and metric. In particular, we analyze the time and temperature behavior of the mean numbers of created particles, provided that the effects of switching the external field on and off are negligible. It is demonstrated that at high temperatures and in slowly varying electric fields the rate of particle-creation is essentially time-dependent.

7. D. M. Gitman, and I.V. Tyutin, Hamiltonization of theories with degenerate coordinates, Nucl. Phys. B630 (3) (2002) pp. 509-527. We consider a class of Lagrangian theories where part of the coordinates does not have any time derivatives in the Lagrange function (we call such coordinates degenerate). We advocate that it is reasonable to reconsider the conventional deﬁnition of singularity based on the usual Hessian and, moreover, to simplify the conventional Hamiltonization procedure. In particular, in such a procedure, it is not necessary to complete the degenerate coordinates with the corresponding conjugate momenta.

8. V.G. Bagrov, D.M. Gitman, A. Levin, and V.B. Tlyachev, Aharonov-Bohm Effect in cyclotron and synchrotron radiations, Nucl. Phys. B605 (2001) 425-454. We study the impact of Aharonov-Bohm solenoid on the radiation of a charged particle moving in a constant uniform magnetic field. With this aim in view, exact solutions of Klein-Gordon and Dirac equations are found in the magnetic-solenoid field. Using such solutions, we calculate exactly all the characteristics of one-photon spontaneous radiation both for spinless and spinning particle. Considering non-relativistic and relativistic approximations, we analyze cyclotron and synchrotron radiations in detail. Radiation peculiarities caused by the presence of the solenoid may be considered as a manifestation of Aharonov-Bohm effect in the radiation. In particular, it

8 is shown that new spectral lines appear in the radiation spectrum. Due to angular distribution peculiarities of the radiation intensity, these lines can in principle be isolated from basic cyclotron and synchrotron radiation spectra.

9. S.P. Gavrilov, D.M. Gitman, Quantization of Point-Like Particles and Consistent Relativistic Quantum Mechanics, Int. J. Mod. Phys. A15 (2000) 4499-4538. We revise the problem of the quantization of relativistic particle models (spinless and spinning), presenting a modified consistent canonical scheme. One of the main point of the modification is related to a principally new realization of the Hilbert space. It allows one not only to include arbitrary backgrounds in the consideration but to get in course of the quantization a consistent relativistic quantum mechanics, which reproduces literally the behavior of the one-particle sector of the corresponding quantum field. In particular, in a physical sector of the Hilbert space a complete positive spectrum of energies of relativistic particles and antiparticles is reproduced, and all state vectors have only positive norms.

10. D.M. Gitman, Path integrals and pseudoclassical description for spinning particles in arbitrary dimensions, Nucl. Phys. B 488 (1997) 490-512. The propagator of a spinning particle in external Abelian field and in arbitrary dimensions is presented by means of a path integral. The problem has distinct solutions in even and odd dimensions. In even dimensions the representation is just a generalization of one in four dimensions (it has been known before). In this case a gauge invariant part of the effective action in the path integral has a form of the standard (Berezin-Marinov) pseudoclassical action. In odd dimensions the solution is presented for the first time and, in particular, it turns out that the gauge invariant part of the effective action differs from the standard one. We propose this new action as a candidate to describe spinning particles in odd dimensions. Studying the hamiltonization of the pseudoclassical theory with the new action we show that the operator quantization leads to ade- quate minimal quantum theory of spinning particles in odd dimensions. Finally the consideration is generalized for the case of the particle with anomalous magnetic moment.

11. S.P. Gavrilov and D.M. Gitman, Vacuum instability in external fields, Phys.Rev.D 53 (1996) 7162-7175. We study particles creation from the vacuum by external electric fields, in particular, by fields, which are acting for a finite time, in the frame of QED in arbitrary space-time dimensions. In all the cases special sets of exact solutions of Dirac equation (IN- and OUT- solutions) are constructed. Using them, characteristics of the effect are calculated. The time and dimensional analysis of the vacuum instability is presented. It is shown that the distributions of particles created by quasi-constant electric fields can be written in a form which has a thermal character and seams to be universal, i.e. is valid for any theory with quasi-constant external fields. Its application, for example, to the particles creation in external constant gravitational field reproduces the Hawking temperature exactly.

12. D.M. Gitman and S.I. Zlatev, Spin factor in path integral representation for Dirac propagator in external field, Phys. Rev. D55 (1997) 7701-7714. We study the spin factor problem both in 3 + 1 and 2 + 1 dimensions which are essentially different for spin factor construction. Doing all Grassmann integrations in the corresponding path integral representations for Dirac propagator we get representations with spin factor in arbitrary external field. Thus, the propagator appears to be presented by means of bosonic path integral only. In 3 + 1 dimensions we present a simple derivation of spin factor avoiding some unnecessary steps in the original brief letter (Gitman, Shvartsman, Phys. Lett. B318 (1993) 122) which themselves need some additional justification. In this way the meaning of the surprising possibility of complete integration over Grassmann variables gets clear. In 2 + 1 dimensions the derivation of the spin factor is completely original. Then we use the representations with spin factor for calculations of the propagator in some configurations of external fields. Namely, in constant uniform electromagnetic field and in its combination with a plane wave field.

13. E.S. Fradkin and D.M. Gitman Path integral representation for the relativistic particle propagators and BFV quantization, Phys. Rev. D 44 (1991) 3230-3236. 9 The path-integral representations for the propagators of scalar and spinor ﬁelds in an external electromagnetic ﬁeld are derived. The Hamiltonian form of such expressions can be interpreted in the sense of Batalin-Fradkin-Vilkovisky quantization of one-particle theory. The Lagrangian representation as derived allows one to extract in a natural way the expressions for the corresponding gauge-invariant (reparametrization- and supergauge-invariant) actions for pointlike scalar and spinning particles. At the same time, the measure and ranges of integrations, admissible gauge conditions, and boundary conditions can be exactly established.

14. E.S. Fradkin and D.M. Gitman, Furry picture for quantum electrodynamics with pair-creating external field, Fortschr. Phys. 29 (1981) 381-411. In the paper the perturbation theory is constructed for QED, for which the interaction with the external pair-creating field is kept exactly. An explicit expression for the perturbation theory causal electron propagator is found. Special features of usage of the unitarity conditions for calculating the total probabilities of radiative processes in the case are discussed. Exact Green functions are introduced and the functional formulation is discussed. Perturbation theory for calculating the mean values of the Heisenberg operators, in particular, of the mean electromagnetic field is built in the case under consideration. Effective Lagrangian which generates the exact equation for the mean electromagnetic field is introduced. Functional representations for the generating functionals introduced in the paper are discussed.

15. D.M. Gitman, Processes of arbitrary order in quantum electrodynamics with a pair-creating external field, Journ. Phys. A 10 (1977) 2007-2020. Dyson’s perturbation theory analogue for quantum electrodynamic processes with arbitrary initial and final states in an external field creating pairs has been discussed. The interaction with the field is taken into account exactly. The possibility of using Feynman diagrams, together with modified correspondence rules, for the representation o the above mentioned processes has been demonstrated.

8 Number of publications and international citations

206 complete articles published in journals. • 26 complete works published in congress annals. • 17 chapters in published books. • 6 published books. • Total of 255 publications. • More than 1700 authentic citations of works in journals and international books (Citation Index). • 9 Coordination of projects and research group Quantization problems and QED in strong fields • FAPESP thematic project 1996/07134-8 (1996-2002) Some current problems in quantum field theory • FAPESP thematic project 2002/00222-9 (2002-2008) Quantization and problems in quantum field theory • FAPESP thematic project 2007/03726-1 (2008-2012) Problems of quantization of non-trivial classical models • CAPES/COFECUB program, No 566/07 (2006-2009) Modern aspects of quantization using coherent states • FAPESP/CNRS program 2009/54771-2 (2010-2012)

10 Coordinator of the research group “Quanta” (Relativistic quantum theory) • Department of Nuclear Physics, Institute of Physics, University of São Paulo. Homepage: http://www.dfn.if.usp.br/pesq/tqr .

10 Summary of the main scientiﬁc results obtained

Obtained results in the following investigation areas:

Quantum ﬁeld theory with external backgrounds • Theory of constrained systems and their quantization • Exact solutions of the relativistic wave equations and theory of self-adjoint extensions • Path integrals; group theory in relativistic quantum mechanics and ﬁeld theory; semiclassical • methods and coherent states

Classical and pseudoclassical relativistic particle models and their quantization • Theory of higher spins • Theory of two and four levels systems and applications to the quantum computation • Quantum mechanics and ﬁeld theory in non-commutative spaces • Quantum Statistics • Other subjects • Below are detailed the results in each area together with the articles where the results were published.

10.1 Quantum field theory with external backgrounds It was elaborated a general formulation of QED with external fields which violate vacuum stability. • In particular, it was constructed a perturbation theory in relation to the radioactive interaction taking into account the interaction with external fields exactly (analogous to the Furry picture in QED with stable vacuum) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 23, 25, 26, 27, 28, 29, 31, 32].

The results obtained for external electromagnetic ﬁelds were generalized for the QFT with non- • abelian and gravitational backgrounds in the articles [33, 34, 16, 35, 20, 36, 37, 38, 56, 47, 61, 58, 57].

Several calculations of particle creation effects were presented. For example, the temporal scenario • of particle creation in electric field [1, 39, 40], calculations in complicated configurations of external fields (combinations of electric, magnetic, and plane wave fields) [2, 41, 42, 43], calculations of the particle creation effects in theories in higher dimensions [40, 47]. It was presented in the articles [44, 45, 46, 28, 62] a general construction of the density matrix of particles created by external fields and it was discovered for the first time a close relationship between the creation of particles in external electromagnetic fields and in gravitational fields [44, 28].

It was calculated the non-perturbative one-loop expression for the mean value of the energy- • momentum tensor of the Dirac field in electric and magnetic field. In this way, the back reaction in external field was obtained, besides the created particles [63, 64, 65, 66].

Radioactive processes were calculated in various electromagnetic ﬁelds [48, 49, 50, 51, 52, 53, 54, • 43, 55, 59, 60, 67].

Some of these results were summed up in the books “Quantum Electrodynamics with Un- • stable Vacuum [30, 31].

11 References

[1] V.G. Bagrov, D.M. Gitman and Sh.M. Shvartsman, Concerning the production of electron-positron pairs from vacuum, Zh. Eksp. Teor. Fiz. 68 (1975) 392-399; Sov. Phys.-JETP, Vol. 41, No. 2 (1975) 191-194.

[2] V.G. Bagrov, S.P. Gavrilov, D.M. Gitman and Sh.M. Shvartsman, Creation of Boson Pairs from Vacuum, Izw. VUZov Fizika 18 No. 3 (1975) 71-74; (Soviet Physics Journal 18 NO.3 (1975) 351-354)

[3] V.G. Bagrov, D.M. Gitman and V.A. Kuchin, External ﬁeld in quantum electrodynamics and coherent states In Actual problems of theoretical physics (Moscow State University Publ., Moscow, 1976) pp. 334-342.

[4] D.M. Gitman, Quantum processes in an intensive electromagnetic ﬁeld. I. Izw. VUZov Fizika (Sov. Phys. Journ.) 10 (1976) 81-86.

[5] D.M. Gitman, Quantum processes in an intensive electromagnetic ﬁeld. II. Izw. VUZov Fizika (Sov. Phys. Journ.) 10 (1976) 86-92.

[6] D.M. Gitman S.P. and Gavrilov, Quantum processes in an intensive electromagnetic ﬁeld creating pairs. III. Izw. VUZov Fizika (Sov. Phys. Journ.) I (1977) 94-99.

[7] D.M. Gitman, Processes of arbitrary order in quantum electrodynamics with a pair-creating external ﬁeld, Journ. Phys. A 10 (1977) 2007-2020.

[8] D.M. Gitman, Processes of arbitrary order in quantum electrodynamics with pair-creating external ﬁeld, In Quantum electrodynamics with external ﬁeld (Tomsk State University, Tomsk, 1977) pp. 132-149.

[9] E.S. Fradkin and D. M. Gitman, Quantum electrodynamics with intense external ﬁeld, Preprint MIT (1978) 1-58.

[10] S.P. Gavrilov, D.M. Gitman and Sh. M. Shvartsman, Green’s functions in external electric ﬁeld, Sov. Journ. Nucl. Phys. (Yadern. Fizika) 29 (1979) 1097-1109

[11] S.P. Gavrilov, D. M. Gitman and Sh.M. Shvartsman, Green’s functions in external electric field and its combination with magnetic field and plane-wave field, Sov. Journ. Nucl. Phys. (Yadern. Fizika), 29 (1979) 1392-1405.

[12] S.P. Gavrilov, D.M. Gitman and Sh.M. Shvartsman, Green’s functions in external electric field and its combination with magnetic field and plane-wave field, Kratk. Soob. Fiz. (Lebedev Inst.), No. 2 (1979) 22-26.

[13] E.S. Fradkin and D.M. Gitman, Problems of quantum electrodynamics with intensive ﬁeld, Preprint PhIAN (Lebedev Institute), 106 (1979) 1-62.

[14] E.S. Fradkin and D.M. Gitman, Problems of quantum electrodynamics with intensive ﬁeld. (Ap- pendix), Preprint PhIAN (Lebedev Institute), 107 (1979) 1-40.

[15] E.S. Fradkin and D.M. Gitman, Problems of quantum electrodynamics with intensive ﬁeld creating pairs, Central research inst. for Phys., Budapest, REKI—1979—83, pp. 1-105.

[16] I.L. Buchbinder and D.M. Gitman, A deﬁnition of the vacuum in curved space-time, Izw. VUZov Fizika (Sov. Phys. Journ.) 7 (1979) 16-21.

[17] S.P. Gavrilov, D.M. Gitman and Sh.M. Shvartsman, The unitarity relation in quantum electrodynamics with pair-creating external ﬁeld, Izw. VUZov Fizika (Sov. Phys. Journ. 257-260) 3 (1980) 93-96 .

[18] S.P. Gavrilov and D.M. Gitman, Furry picture for scalar quantum electrodynamics with intensive pair-creating ﬁeld, Izw. VUZov Fizika (Sov. Phys. Journ. 491-496) 6 (1980) 37-42 . 12 [19] E.S. Fradkin and D.M. Gitman, Furry picture for quantum electrodynamics with pair-creating external ﬁeld, Fortschr. Phys. 29 (1981) 381-411.

[20] I.L. Buchbinder, E.S. Fradkin and D.M. Gitman, Generating functional in quantum ﬁeld theory with unstable vacuum, Preprint PhIAN (Lebedev Institute), 138 (1981).

[21] D.M. Gitman and V.A. Kuchin, Generating functional of mean ﬁeld in quantum electrodynamics with unstable vacuum, Izw. VUZov Fizika (Sov. Phys. Journ.) 10 (1981) 80-84.

[22] S.P. Gavrilov, D.M. Gitman and E.S. Fradkin, Quantum electrodynamics at ﬁnite temperature in presence of an external ﬁeld, violating the vacuum stability, Sov. Journal Nucl. Phys. (Yadernaja Fizika), 46 (1987) 172-180.

[23] E.S. Fradkin, D.M. Gitman and Sh.M. Shvartsman, Optical theorem in quantum electrodynamics with unstable vacuum, Fortschr. Phys. 36 (1988) 643-669.

[24] V.G. Bagrov, V.P. Barashev, D.M. Gitman, and Sh.M. Shvartsman, Green functions in external electromagnetic ﬁeld, in Collection Quantum processes in intense external ﬁelds, pp. 101-111, (Shteentza, Kishenev, 1987) (3B348b88)

[25] V.P. Barashev, D.M. Gitman, E.S. Fradkin and Sh.M. Shvartsman, Peculiarities of reduction formulas in quantum electrodynamics with unstable vacuum, Preprint PhIAN (Lebedev Institute) 177 (1988) 1-26.

[26] D.M. Gitman, E.S. Fradkin and Sh.M. Shvartsman, Quantum electrodynamics with external ﬁeld, violating the vacuum stability, Trudy PhIAN (Proceedings of Lebedev Institute, Moscow), 193 (1989) 3-207.

[27] S.P. Gavrilov, D.M. Gitman and E.S. Fradkin, Quantum electrodynamics at ﬁnite temperature in presence of an external ﬁeld, violating the vacuum Stability, Trudy PhIAN (Proceedings of Lebedev Institute, Moscow), 193 (1989) 208-221.

[28] V.P. Barashev, E.S. Fradkin, D.M. Gitman and Sh.M. Shvartsman, The problems of QED with unstable vacuum. Reduction formulas. The density matrix of particles creating in an external ﬁeld, Trudu PhIAN (Proceedings of Lebedev Institute, Moscow) 201 (1990) 74-94.

[29] S.P. Gavrilov and D.M. Gitman, Interpretation of external ﬁeld and external current in QED, Sov. Journ. Nucl. Phys. (Yadern. Fizika) 51 (1990) 1644-1654.

[30] E.S. Fradkin, D.M. Gitman and Sh.M. Shvartsman, Quantum Electrodynamics with Unsta- ble Vacuum (Springer-Verlag, Berlin Heidelberg New-York London Paris Hong-Kong Barcelona, 1991) pp. 1-300.

[31] D.M. Gitman, E.S. Fradkin and Sh.M. Shvartsman, Quantum Electrodynamics with Unstable Vacuum, (Nauka, Moscow, 1991) pp. 1—294.

[32] S.P.Gavrilov, D.M.Gitman, Furry Representation for Fermions, interacting with an external gauge ﬁeld, Izw. VUZov Fizika (Russian Phys. Journ.) No 4 (1995) 102-108.

[33] I.L. Buchbinder and D.M. Gitman, A method of calculation of quantum processes probabilities in external gravitational ﬁelds. I, Izw. VUZov Fizika (Sov. Phys. Journ.) 3 (1979) 90-95.

[34] I.L. Buchbinder and D.M. Gitman, A method of calculation of quantum processes probabilities in external gravitational ﬁelds. II, Izw. VUZov Fizika (Sov. Phys. Journ.) 4 (1979) 55-61.

[35] I.L. Buchbinder, E.S. Fradkin and D.M. Gitman, Quantum electrodynamics in curved space-time, Fortschr. Phys. 29 (1981) 187-218.

[36] I.L. Buchbinder, E.S. Fradkin and D.M. Gitman, Quantum electrodynamics in curved space-time, Trudu PhIAN (Proceedings of Lebedev Institute, Moscow) 201 (1990) 33—73.

13 [37] S.P. Gavrilov and D.M. Gitman, Problems of an External Field in Non-Abelian Gauge Theory on an example of the standard SU(2) U(1) model, Preprint MIT, CTP # 1995 (1991) 1-53; Problems × of an External Field in Non-Abelian Gauge Theory, Proceedings of the First International Sakharov Conference on Physics, “Sakharov Memorial Lectures in Physics” Vol.2 (1991) 187-194, Edited by L.V. Keldysh and V.Ya. Fainberg, Nova Science Publishers, Inc.

[38] S.P.Gavrilov, D.M.Gitman, Green’s Functions and Matrix Elements in Furry Picture for Elec- troweak Theory with non-Abelian External Field, Izw. VUZov Fizika (Russian Phys. Journ.) 36, No 5 (1993) 448-452.

[39] D.M. Gitman, V.M. Shachmatov and Sh.M. Shvartsman, Pair creation in the electric ﬁeld, acting for a ﬁnite time, Izw. VUZov Fizika (Sov. Phys. Journ.) 4 (1975) 23-29.

[40] S.P. Gavrilov and D.M. Gitman, Vacuum instability in external ﬁelds, Phys. Rev. D 53 (1996) 7162-7175

[41] V.G. Bagrov, D.M. Gitman and Sh.M. Shvartsman, Pair creation from vacuum by an electromagnetic ﬁeld in the zero-plane formalism, Sov. Journ. Nucl. Phys. (Yadern. Fizika), 23 (1976) 394-400.

[42] S.P. Gavrilov and D.M. Gitman, Processes of pair-creation and scattering in constant ﬁeld and plane-wave ﬁeld, Izw. VUZov Fizika (Sov. Phys. Journ.) 5 (1981) 108-111.

[43] D.M. Gitman, M.D. Noskov and Sh.M. Shvartsman, Quantum effects in a combination of a constant uniform field and a plane wave field, Intern. Journ. Mod. Phys. A 6 (1991) 4437—4489.

[44] V.P. Frolov and D.M. Gitman, Density matrix in quantum electrodynamics, equivalence principle and Hawking eﬀect, Journ. Phys. A 15 (1978) 1329-1333.

[45] D.M. Gitman V.P. and Frolov, Density matrix in quantum electrodynamics and Hawking eﬀect, Sov. Journ. Nucl. Phys. (Yadern. Fizika), 28 (1978) 552-557.

[46] I.L. Buchbinder, D.M. Gitman and V.P. Frolov, Density matrix for particle-creation processes in external ﬁeld, Izw. VUZov Fizika (Sov. Phys. Journ. 529-533) 6 (1980) 77-81.

[47] S.P. Gavrilov, D.M. Gitman, and A.E. Gonçalves, QED in external ﬁeld with space-time uniform invariants: Exact solutions, Journ. Math. Phys. 39 (1998) 3547-3567

[48] V.G. Bagrov, P.V. Bozrikov, D.M. Gitman, Yu.I. Klimenko and A.I. Khudomjasov, Radiation of Neutral Fermion with Electric and Magnetic Moments in Constant and Uniform External Electro- magnetic Fields, Izw. VUZov Fizika 17 No. 6 (1974) 150-151; (Soviet Physics Journal 17 No. 6 (1974) 890-891)

[49] V.G. Bagrov, P.V. Bozrikov, D.M. Gitman, Yu.I. Klimenko and A.I. Khudomjasov, Electromagnetic Wave Scattering at a Neutral Fermion Possessing Magnetic and Electric Moments, Izw. VUZov Fizika 17 No. 7 (1974) 138-139; (Soviet Physics Journal 17 No. 7(1974) 1072-1028)

[50] V.G. Bagrov, D.M. Gitman, A.A. Sokolov et al., Radiation of relativistic electrons, moving in the ﬁnite length ondulator, Journ. Technic. Fiz. XLV, 9 (1975) 1948-1953.

[51] V.G. Bagrov, D.M. Gitman and V.N. Rodionov et al., Eﬀect of a strong electromagnetic wave on the radiation emitted by weakly excited electrons, moving in magnetic ﬁeld, Zh. Eksp. Teor. Fiz. 71 (1976) 433-439.

[52] Yu.Yu. Volfengaut, S.P. Gavrilov, D.M. Gitman and Sh.M. Shvartsman, Radiative processes in external pair-creating electromagnetic ﬁeld, Sov. Journ. Nucl. Phys. (Yadern. Fizika) 33 (1981) 743-757.

[53] S.P. Gavrilov and D.M. Gitman, Vacuum radiative processes in pair-creating ﬁelds, Izw. VUZov Fiz. 9 (1982) 10-12 (Sov. Phys. Journ. 9 (1982) 775-778.)

14 [54] S.P. Gavrilov and D.M. Gitman, Radiative processes with an electron in constant homogeneous ﬁeld , Izw. VUZov Fiz. 10 (1982) 102-106 (Sov. Phys. Journ. 10 (1982) 968-972.)

[55] D.M. Gitman, S.D. Odintsov and Yu.I. Shil’nov, Chiral symmetry breaking in d = 3 NJL model in external gravitational and magnetic ﬁelds, Phys. Rev. D 54, No 4 (1996) 2968-2970

[56] S.P. Gavrilov, D.M. Gitman and S.D. Odintsov, Quantum Scalar Fields in the FRW Universe with a Constant Electromagnetic Background, Int. J. Mod. Phys. A12 (1997) 4837-4867

[57] S.P. Gavrilov and D.M. Gitman, Quantum processes in FRW Universe with external electromagnetic ﬁeld, Proceedings 8th Lomonosov Conference on Elementary Particle Physics (25-30 August 1997, Moscow, Russia), Ed. A.I. Studenikin, (Int. Centre for Advanced Studies, Moscow 1999) 105-109

[58] S.P. Gavrilov, D.M. Gitman, and A.E. Gonçalves, Quantum Spinor Field in FRW Universe with Constant Electromagnetic Background, Int.J.Mod.Phys.A16, No.26 (2001) 4235-4259

[59] I. Brevik, D.M. Gitman and S.D. Odintsov, The eﬀective potential of gauged NJL model in a magnetic ﬁeld, Gravitation and Cosmology, 3 (1997) 100-104

[60] I. Brevik, D.M. Gitman, and S.D. Odintsov, The Eﬀective Potential of Gauged NJL Model in Magnetic Field, in Proceedings of 1996 International Workshop PERSPECTIVES OF STRONG COUPLING GAUGE THEORIES, (Nagoya, 13-16 November 1996, Japan), Editors J. Nishimura and K. Yamawaki, (World Sci. Singapore, 1997) pp. 208-214

[61] S.P. Gavrilov, D.M. Gitman, The Proper-Time representation of Spinor Green Functions in FRW Universe with Electromagnetic Background and some Applications of Them, Proceedings of Forth Alexander Friedmann International Seminar on Gravitation and Cosmology, St. Petersburg, Rus- sia, June 17-25, 1998/editors: Yu.N. Gnedin [et al]- Campinas, SP: UNICAMP/IMECC, 1999, pp.268-273

[62] S.P. Gavrilov, D.M. Gitman, and J.L. Tomazelli, Density matrix of a quantum ﬁeld in a particle- creating background, Nucl. Phys. B 795 [FS] (2008) 645-677

[63] S.P. Gavrilov, D.M. Gitman, One-loop energy-momentum tensor in QED with electric-like background, Phys. Rev. D78, 045017(35) (2008)

[64] S.P. Gavrilov and D.M. Gitman, Energy-momentum tensor in thermal strong-ﬁeld QED with unstable vacuum, arXiv:0710.3933; J. Phys. A: Math. Theor. 41 (2008) 164046.

[65] S.P. Gavrilov and D.M. Gitman, Consistency Restrictions on Maximal Electric-Field Strength in Quantum Field Theory, Phys. Rev. Lett. 101, 130403(4) (2008)

[66] S.P. Gavrilov, D.M. Gitman, On Schwinger Mechanism for Gluon Pair Production in the Presence of Arbitrary Time Dependent Chromo-Electric Field, Europ. Physical Journal C, 64, Issue 1 (2009) 81; DOI: 10.1140/epjc/s10052-009-1135-7

[67] M. Bordag, I.V. Fialkovsky, D.M. Gitman, and D.V. Vassilevich, Casimir interaction between a perfect conductor and graphene, described by the Dirac model, Phys. Rev. B 80, No.24 (2009)

10.2 General theory of constrained systems and their quantization The structure of the physical sector of gauge theories, in general form, was exhaustively described • in the Lagrangian and Hamiltonian formulations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. For example, it was proved for the first time that the number of gauge transformations of the action is equal to the number of primary first class constraints in the Hamiltonian formulation and that the number of non-physical variables is equal to the number of first class constraints.

The Hamiltonization and quantization of singular systems with higher derivatives was formulated • for the ﬁrst time [12, 13].

15 It was formulated also for the first time the Hamiltonization and canonical quantization of sys- • tems with time-dependent constraints and the method was applied to the quantization of zero Hamiltonian systems [14, 15, 16, 17, 18, 19]. All the symmetries of a general gauge theory were described and, in particular, the form of • gauge transformation was related to the constraint structure of the same theory in Hamiltonian formulation. In this way, it was obtained a rigorous proof of the equivalence of the two definitions of physical quantities in gauge theories: one of them states that the physical functions are gauge invariants in the extremals and the other states that the physical functions are those which commute with first class constraints (Dirac conjecture) [20, 21, 22, 23, 24, 25, 26, 27]. The so called triplectic quantization of gauge theories was developed together with an extension • of the BRST—anti BRST superquantization scheme which is covariant in general coordinates [28, 29, 30, 31, 32, 33, 34, 35, 36]. It was developed a method to quantize systems with non-Lagrangian movement equations [37, • 38, 39, 40]. The main results were published in the books Canonical Quantization of Fields with Con- • straints [14], and Quantization of Fields with Constraints [15].

References

[1] D.M. Gitman and I.V. Tyutin, Canonical quantization of singular theories, Izw. VUZov Fizika (Sov. Phys. Journ. 423-439) 5 (1983) 3-22. [2] D.M. Gitman, Ya.S. Prager and I.V. Tyutin, Special canonical coordinates in constrained theories, Izw. VUZov Fizika (Sov. Phys. Journ. 760-764) 8 (1983) 93-97. [3] D.M. Gitman, S.L. Ljachovich, M.D. Noskov and I.V. Tyutin, Lagrangian formulation of Hamil- tonian theory of general form with constraints, Proceedings of III Urmala Seminar, Urmala, 1985, v. 2 (Nauka, Moscow 1986) pp. 316-322. [4] D.M. Gitman and I.V. Tyutin, Canonical quantization of gauge theories of special form, Izw. VUZov Fizika (Sov. Phys. Journ.) 3 (1986) 176-187. [5] D.M. Gitman, S.L. Ljachovich, M.D. Noskov and I.V. Tyutin, Lagrangian formulation of Hamil- tonian theory of general form with constraints, Izw. VUZov Fizika (Sov. Phys. Journ.) 3 (1986) 243-250. [6] D.M. Gitman and I.V. Tyutin, The structure of gauge theories in the Lagrangian and Hamiltonian formalisms, In Quantum field theory and quantum statistics v. I, pp. 143—164, Ed. by Batalin, Isham and Vilkovisky (Adam Hilger, Bristol, 1987). [7] D.M. Gitman and I.V. Tyutin, Canonical quantization of singular theories, In Group theoretical methods in physics Proceedings of Second Zvenigorod Seminar Group Theoretical Methods Physics, Zvenigorod, USSR, 1988, pp. 207-237. [8] D.M. Gitman, P.M. Lavrov and I.V. Tyutin, Non-point transformation in constrained theories, Journal Phys. A 23 (1990) 41-51. [9] D.M. Gitman, Canonical and D-transformations in theories with constraints, Int.J.Theor.Phys. 35, No.1 (1996) 87-99 [10] D. M. Gitman, and I.V. Tyutin, Hamiltonization of theories with degenerate coordinates, Nucl. Phys. B630 (3) (2002) pp. 509-527 [11] D. M. Gitman, and I.V. Tyutin, Hamiltonian Formulation of Theories with Degenerate Coordi- nates, Proceedings of 3-rd International Sakharov Conference on Physics, Moscow, Russia, June 24-29, 2002, Vol.II, Editors A. Semikhatov, M. Vasiliev, V. Zaikin (Scientific World Publ. 2003) pp. 54-63 16 [12] D.M. Gitman, S.L. Ljachovich and I.V. Tyutin, Hamiltonian formalism of theories with higher derivatives, Izw. VUZov Fizika (Sov. Phys. Journ. 730-735) 8 (1983) 61-66. [13] D.M. Gitman, S.L. Ljachovich and I.V. Tyutin, Canonical quantization of Yang-Mills theory with higher derivatives, Izw. VUZov Fizika (Sov. Phys. Journ.) 7 (1985) 37-40. [14] D.M. Gitman and I.V. Tyutin, Canonical quantization of fields with constraints (Nauka, Moscow, 1986) pp. 1-216. [15] D.M. Gitman and I.V. Tyutin, Quantization of Fields with Constraints pp. 1—291 (Springer- Verlag, Berlin Heidelberg New-York London Paris Hong-Kong Barcelona, 1990). [16] S.P. Gavrilov and D.M. Gitman, Quantization of Systems with Time-Dependent Constraints. Ex- ample of Relativistic Particle in Plane Wave, Class. Quantum Grav. 10 (1993) 57-67. [17] G. Fulop, D.M. Gitman and I.V. Tyutin, Reparametrization Invariance as Gauge Symmetry, Int. J. Theor. Phys. 38 (1999) 1953-1980 [18] G. Fulop, D.M. Gitman, and I.V. Tyutin, Reparametrization Invariance and Zero Hamiltonian Phenomenon, Proceedings 8th Lomonosov Conference on Elementary Particle Physics (25-30 August 1997, Moscow, Russia), Ed. A.I. Studenikin, (Int. Centre for Advanced Studies, Moscow 1999) 64-69 [19] G. Fulop, D.M. Gitman, and I.V. Tyutin, Reparametrization Invariance and Zero Hamiltonian Phenomenon, in Topics in Theoretical Physics II, Festschrift for Abraham H. Zimerman Ed. H. Aratyn, L.Ferreira, J. Gomes, (IFT/UNESP-São Paulo-SP-Brazil-1998) pp. 286-295 [20] D. M. Gitman, and I.V. Tyutin, Constraint reorganization consistent with Dirac procedure, Michael Marinov Memorial Volume: Multiple Facets of Quantization and Supersymmetry, ed. M. Olshanetsky and A. Vainstein (World Publishing, Singapore 2002) pp.184-204 [21] D. M. Gitman, and I.V. Tyutin, Fine Structure of Constraints in Hamiltonian Formulation, Grav- itation & Cosmology, 8, No.1-2 (2002) 138-140 [22] B. Geyer, D.M. Gitman, and I.V. Tyutin, Canonical form of Euler-Lagrange equations and gauge symmetries, J. Phys. A36 (2003) 6587-6609 [23] B. Geyer, D.M. Gitman, and I.V. Tyutin, Reduction of Euler-Lagrange equations in general gauge theories with external fields, Proceedings of Sixth Workshop (The University of Oklahoma, Norman, OK USA September 15-19, 2003) on QUANTUM FIELD THEORY UNDER THE INFLUENCE OF EXTERNAL CONDITIONS, Ed. K.A. Milton, (Rinton Press 2004) pp.276 - 281. [24] D.M. Gitman, I.V. Tyutin, General quadratic gauge theory. Constraint structure, symmetries, and physical functions, J. Phys. A: Math. Gen. 38 (2005) 5581-5602. [25] D.M. Gitman, I.V. Tyutin, Symmetries in Constrained Systems, Resenhas IME-USP, Vol. 6, U 2116 2/3 (2004) pp. 187-198 [26] D.M. Gitman, I.V. Tyutin, Symmetries and physical functions in general gauge theory, Int. J. Mod. Phys.A, 21, No.2 (2006) pp. 327-360 [27] D.M. Gitman, I.V. Tyutin, Symmetries of Dynamically Equivalent Theories, Brazilian Journal of Physics, 36, no.1A (2006) pp. 132-140 [28] B. Geyer, D.M. Gitman, and P.M. Lavrov A modified scheme of triplectic quantization, Mod. Phys. Lett. A14 (1999) pp. 661-670 [29] B. Geyer, D.M. Gitman, and P.M. Lavrov, Triplectic quantization of gauge theories, Theor. Math. Phys. 123, No.3 (2000) 813-820 [30] B. Geyer, D.M. Gitman, and P.M. Lavrov, Covariant Quantization with Extended BRST Symme- try, Proceedings of International Seminar Physical Variables in Gauge Theories, Dubna,September 21-25, Russia, 1999, Ed. by A.Khvedelidze, M.Lavelle, D.McMullan, and V.Pervushin, Dubna, 2000, pp.118-128 17 [31] B. Geyer, D.M. Gitman, P. Lavrov, P. Moshin, On Problems of the Lagrangian Quantization of W3-gravity, Int.J.Mod.Phys. A18, No.27 (2003) 5099-5125

[32] B. Geyer, D.M. Gitman, P. Lavrov, P. Moshin, Superﬁeld Extended BRST Quantization in General Coordinates, Int.J.Mod.Phys.A19 (2004) pp.737-750

[33] D.M. Gitman, P.Yu. Moshin, J.L. Tomazelli, On superﬁeld covariant quantization in general coordinates, Eur. Phys. J. C44 (2005) 591-598

[34] D.M. Gitman, P.Yu. Moshin, A.A. Reshetnyak, Local Superﬁeld Lagrangian BRST Quantization, J. Math. Phys. 46:072302 (2005)

[35] D.M. Gitman, P.Yu. Moshin, and A.A. Reshetnyak, An embedding of the BV quantization into an N=1 local superﬁeld formalism, Publicação IFUSP - 1609/2005, hep-th/0507046, Phys. Lett.B 621 (2005) pp. 295-308

[36] D.M. Gitman , P.Yu. Moshin, A.A. Reshetnyak, Reducible gauge theories in local superﬁeld La- grangian BRST quantization, Brazilian Journal of Physics, 37 (2007) no. 4

[37] D. Gitman and V.G. Kupriyanov, Canonical quantization of non-Lagrangian theories and its application to damped oscillator and radiating point-like charge, Eur. Phys. J. C50 (2007) 691-700

[38] D. Gitman and V.G. Kupriyanov, Quantization of Theories with non-Lagrangian Equations of Motion, Journal of Math. Sciences 141 (2007) 1399-1406

[39] D.M. Gitman, V.G. Kupriyanov, Action principle for so-called non-Lagrangian systems, PoS (IC2006) 016 (2006) pp 1-11

[40] D.M. Gitman and V.G. Kupriyanov, The action principle for a system of diﬀerential equations, J. Phys. A: Math. Theor. 40 (2007) 10071-10081.

10.3 Exact solutions of the relativistic wave equations and theory of self-adjoint extensions It was obtained and systematically investigated for the ﬁrst time a large number of new classes • of exact solutions for relativistic wave equations in external ﬁelds [1, 2, 3, 4, 5, 6, 7, 3, 11, 8, 9, 10, 11, 12, 13, 14, 15, 16, 13, 17, 21, 23, 24, 22, 18, 19, 20].

It was systematically studied a new exact QED model (electron interacting with a plane wave) • [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 11, 12, 34].

Various species of Green functions for relativistic wave equations were calculated with diﬀerent • methods, for example through the set of exact solutions, Schwinger’s proper time method and path integral method [35, 36, 37, 38, 39, 40, 42, 41, 43, 44, 45, 46, 47].

Some of the results of the above activities are summed up in the books Exact Solutions of • Relativistic Wave Equations [21, 27].

New exact solutions for the relativistic wave equations in 3+1 and 2+1 dimensions in combination • with solenoidal Aharonov-Bohm field and some additional electric and magnetic fields were studied in detail [48, 49, 50, 51, 52, 53, 54]. The obtained solutions were used in the study of the Aharonov- Bohm effect under the correspondent electromagnetic fields [55, 56, 57, 58, 59, 60, 61, 62, 63].

It was developed an adaptation of the general theory of the self-adjoint extensions for the use in • physical problems with various applications [64, 65, 66, 67, 68, 69, 70].

18 References

[1] V.G. Bagrov, D.M. Gitman, V.N. Zadorozhny and Sh.M. Shvartsman, An Electron in the Field of Two Classical Plane Waves Propagating in Slightly Diﬀerent Directions, Soviet Physics Journal 18 No. 1 (1975) 34-36)

[2] V.G. Bagrov, D.M. Gitman, P.M. Lavrov, V.N. Zadorozhny and V.N. Shapovalov, New exact solutions of the Dirac equation. II., Sov. Phys. Journ. 4 (1975) 29-33.

[3] V.G. Bagrov, D.M. Gitman, P.M. Lavrov, V.N. Zadorozhny and V.N. Shapovalov, New exact solutions of the Dirac equation. III., Sov. Phys. Journ. 7 (1975) 7-11.

[4] V.G. Bagrov, D.M. Gitman, A.G. Meshkov et al., New exact solutions of the Dirac equation. IV. Izw. VUZov Fizika (Sov. Phys. Journ.) 8 (1975) 73-79.

[5] D.M. Gitman, V.M. Shachmatov and Sh.M. Shvartsman, Completeness and orthogonality on the null-plane of one class of solutions of relativistic wave equations, Izw. VUZov Fizika (Sov. Phys. Journ.) 8 (1975) 43-49.

[6] V.G. Bagrov, N.N. Bizov, D.M. Gitman et al., New exact solutions of the Dirac equation. V. Izw. VUZov Fizika (Sov. Phys. Journ.) 9 (1975) 105-111.

[7] V.G. Bagrov, D.M. Gitman and A.V. Jushin, Solutions for the motion of an electron in electromagnetic ﬁeld, Phys. Rev. D 12 (1975) 3200-3201.

[8] V.G. Bagrov, D.M. Gitman, A.G. Meshkov and V. N. Shapovalov, Supplement to the work New exact solutions of the Dirac equation. II, III Izw. VUZov Fizika (Sov. Phys. Journ.) 1 (1977) 126-127.

[9] V.G. Bagrov, D.M. Gitman, A.V. Shapovalov and V.N. Shapovalov, New exact solutions of the Dirac equation. VI Izw. VUZov Fizika (Sov. Phys. Journ.) 6 (1977) 105-114.

[10] V.G. Bagrov, D.M. Gitman, N.B. Suchomlin et al., New exact solutions of the Dirac equation. VII. Izw. VUZov Fizika (Sov. Phys. Journ.) 7 (1977) 46-51.

[11] V.G. Bagrov, D.M. Gitman, P.M. Lavrov and V.N. Zadorozhni, Characteristic features of exact solutions of the problem of an electron in quantized ﬁeld of a plane wave, Izw. VUZov Fizika (Sov. Phys. Journ.) 3 (1977) 7-14.

[12] V.G. Bagrov, D.M. Gitman and A. V. Shapovalov, Integrals of motion in the electron in quantized plane-wave problem, Izw. VUZov Fizika (Sov. Phys. Journ.) 2 (1977) 116-121.

[13] V.G. Bagrov and D.M. Gitman, Exact solutions of the relativistic wave-equations in external ﬁeld, In Quantum electrodynamics with external ﬁelds, (Tomsk State University, Tomsk, 1977) pp. 5-100.

[14] V.G. Bagrov, D.M. Gitman, V.N. Zadorozhni et al., New exact solutions of the Dirac equation. VIII, Izw. VUZov Fizika (Sov. Phys. Journ.) 2 (1978) 13-18.

[15] V.G. Bagrov, D.M. Gitman, V.N. Zadorozhni et al., New exact solutions of the Dirac equation. IX, Izw. VUZov Fizika (Sov. Phys. Journ.) 3 (1978) 46-49.

[16] V.G. Bagrov, D.M. Gitman, V.N. Zadorozhni at al.: New exact solutions of the Dirac equation, Izw. VUZov Fizika (Sov. Phys. Journ. 276-281) 4 (1980) 10-16.

[17] V.G. Bagrov, D.M. Gitman and V.N. Shapovalov, Electron motion in longitudinal electromagnetic ﬁelds, J. Math. Phys. 23 (1982) 2558-2561.

[18] S.P. Gavrilov, D.M. Gitman, and J.L. Tomazelli, Comments on spin operators and spin-polarization states of 2 + 1 fermions, Eur. Phys. J. C (2005) DOI: 10.1140/epjc/s2004-02026-9

[19] V.G. Bagrov, D.M. Gitman, Non-Volkov solutions for a charge in a plane wave, Annalen der Physik 14, 8 (2005) pp. 467-478

19 [20] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, Charged particles in crossed and longitudinal electromagnetic ﬁelds and beam guides, J. Math. Phys., 48, 8 (2007) 082305-1, ..., 082305-15

[21] V.G. Bagrov, D.M. Gitman, I.M. Ternov et al. Exact Solutions of the Relativistic Wave Equations (Nauka, Novosibirsk, 1982) pp. 1-144.

[22] V.G. Bagrov and D.M. Gitman, Exact Solutions of Relativistic Wave Equations pp. 1—321 (Kluwer Acad. Publisher, Dordrecht Boston London, 1990).

[23] V.G. Bagrov, P.V. Bozrikov and D.M. Gitman, A charge in quantized plane-wave ﬁeld, Izw. VUZov Radioﬁzika (Sov. Journ. Radiophys.), XVI, I (1973) 129-140.

[24] V.G. Bagrov, P.V. Bozrikov and D.M. Gitman, An Electron in the quantized electromagnetic plane- wave ﬁeld, Theor. Mat. Fiz., 14 2 (1973) 202-210.

[25] V.G. Bagrov, P.V. Bozrikov, D.M. Gitman and P.M. Lavrov, An Electron in a Field of a Plane Quantized Monochromatic Electromagnetic Wave, Izw. VUZov Fizika 16 No. 8 (1973) 55-58; (Soviet Physics Journal 16 No. 8 (1973) 1082-1085)

[26] V.G. Bagrov, D.M. Gitman, and V.A. Kuchin, Interaction with an external ﬁeld in quantum electrodynamics, Izw. VUZov Fizika (Sov. Phys. Journ.) 4 (1974) 152-153.

[27] V.G. Bagrov, D.M. Gitman and P.M. Lavrov, Electron in a Quantized Field of a Plane Wave and in a Classical Field of Redmond’s Conﬁguration, Izw. VUZov Fizika 17 No.6 (1974) 47-51; (Soviet Physics Journal 17 No. 6 (1974) 787-790)

[28] V.G. Bagrov, D.M. Gitman and P.M. Lavrov, Electron in Constant Crossed Electromagnetic Fields and Plane-Wave Fields, Izw. VUZov Fizika 17 NO.6 (1974) 68-74; (Soviet Physics Journal 17 No.6 (1974) 806-811)

[29] V.G. Bagrov, P.V. Bozrikov and D.M. Gitman, Fermion with Anomalous Moment in a Field of Quantized Plane Wave, Izw. VUZov Fizika 17 No. 6 (1974) 129-132; (Soviet Physics Journal 17 No. 6(1974) 864-866)

[30] V.G. Bagrov, D.M. Gitman and V.A. Kuchin, Electron in the Field of Classical and a Quantized Plane Wave traveling in the Same Direction, Izw. VUZov Fizika 17 No. 7 (1974) 60-64; (Soviet Physics Journal 17 No. 7 (1974) 952-956)

[31] V.G. Bagrov, D.M. Gitman, V.A. Kuchin and P.M. Lavrov, Bases of electrodynamics of electrons interacting with quantized plane wave ﬁeld. I, Izw. VUZov Fizika (Sov. Phys. Journ.) 12 (1974) 89-94.

[32] V.G. Bagrov, D.M. Gitman, V.A. Kuchin and P.M. Lavrov, Bases of electrodynamics of electrons, interacting with quantized plane wave ﬁeld. II, Izw. VUZov Fizika (Sov. Phys. Journ.) 7 (1975) 11-15.

[33] V.G. Bagrov, D.M. Gitman and Sh.M. Shvartsman, Electron in a Quantized Plane Wave-Field and the Classical Field of a Longitudinal Electric Wave, Izw. VUZov Fizika 18 No. 3 (1975) 67-71; (Soviet Physics Journal 18 No. 3 (1975) 374-350)

[34] V.G. Bagrov, I.L. Buchbinder, D.M. Gitman and P.M. Lavrov, Coherent states of the electron in quantized electromagnetic wave, Theor. Mat. Fiz. 33 (1977) 419-426.

[35] S.P. Gavrilov, D.M. Gitman and Sh. M. Shvartsman, Green’s functions in external electric ﬁeld, Sov. Journ. Nucl. Phys. (Yadern. Fizika) 29 (1979) 1097-1109

[36] S.P. Gavrilov, D. M. Gitman and Sh.M. Shvartsman, Green’s functions in external electric field and its combination with magnetic field and plane-wave field, Sov. Journ. Nucl. Phys. (Yadern. Fizika), 29 (1979) 1392-1405.

[37] S.P. Gavrilov, D.M. Gitman and Sh.M. Shvartsman, Green’s functions in external electric field and its combination with magnetic field and plane-wave field, Kratk. Soob. Fiz. (Lebedev Inst.), No. 2 (1979) 22-26. 20 [38] E.S. Fradkin and D.M. Gitman, Problems of quantum electrodynamics with intensive field. (Ap- pendix), Preprint PhIAN (Lebedev Institute), 107 (1979) 1-40.

[39] V.G. Bagrov, V.P. Barashev, D.M. Gitman, and Sh.M. Shvartsman, Green functions in external electromagnetic ﬁeld, in Collection Quantum processes in intense external ﬁelds, pp. 101-111, (Shteentza, Kishenev, 1987) (3B348b88)

[40] D.M. Gitman, M.D. Noskov and Sh.M. Shvartsman, Green’s functions in external electromagnetic ﬁeld, Izw. VUZov Fizika (Sov. Phys. Journ.) 5 (1989) 59-64.

[41] D.M. Gitman, M.D. Noskov and Sh.M. Shvartsman, Quantum effects in a combination of a constant uniform field and a plane wave field, Intern. Journ. Mod. Phys. A 6 (1991) 4437—4489.

[42] S.P.Gavrilov, D.M.Gitman, Green’s Functions and Matrix Elements in Furry Picture for Elec- troweak Theory with non-Abelian External Field, Izw. VUZov Fizika (Russian Phys. Journ.) 36, No 5 (1993) 448-452.

[43] D.M. Gitman, Sh.M.Shvartsman and W.da Cruz, Path Integral over Velocities for Relativistic Particle Propagator, Bras. Journ. Phys. 24, No.4 (1994) 844-854.

[44] D.M. Gitman, S.I. Zlatev and W.da Cruz, Spin Factor and Spinor Structure of Dirac Propagator in Constant Field, Bras. Journ. Phys. 26 (1996) 419-425

[45] D.M. Gitman and S.I. Zlatev, Spin factor in path integral representation for Dirac propagator in external ﬁeld, Phys. Rev. D55 (1997) 7701-7714

[46] S.P. Gavrilov and D.M. Gitman, Proper time and path integral representations for the commutation function, J. Math. Phys. 37 (7) (1996) 3118-3130

[47] D.M. Gitman and S.I. Zlatev, Semiclassical Form of the Relativistic Particle Propagator, Mod. Phys. Lett.A 12 (1997) 2435-2443

[48] V.G. Bagrov, D.M. Gitman, and V.B. Tlyachev, The exact solutions of relativistic wave equations for Aharonov-Bohm ﬁeld in combination with other electromagnetic ﬁelds, Proceedings of FORA, No. 6 (2001) 11-4

[49] V.G. Bagrov, D.M. Gitman, and V.B. Tlyachev, Solutions of relativistic wave equations in super- positions of Aharonov-Bohm, magnetic, and electric ﬁelds, J. Math. Phys. 42, No.5 (2001)

[50] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, and I.V. Shirokov, New solutions of relativistic wave equations in magnetic ﬁeld and longitudinal ﬁelds, J.Math. Phys. 43 (2002) 2284-2295

[51] S.P. Gavrilov, D.M. Gitman, and A.A. Smirnov, Dirac equation in the magnetic-solenoid ﬁeld, Europ. Phys. Journ. C 30 (2003) 009

[52] S.P. Gavrilov, D.M. Gitman, and A.A. Smirnov, Green functions of the Dirac equation with magnetic-solenoid ﬁeld, J. Math. Phys. 45 (2004) 1873

[53] S.P. Gavrilov, D.M. Gitman, and A.A. Smirnov, Self-adjoint extensions of Dirac Hamiltonian in magnetic-solenoid ﬁeld and related exact solutions, Phys. Rev. A 67 (2003) 024103(4)

[54] S.P. Gavrilov, D.M. Gitman, A.A. Smirnov, and B.L. Voronov, Dirac fermions in a magnetic- solenoid ﬁeld, ”Focus on Mathematical Physics Research” Ed. by Charles V. Benton (Nova Science Publishers, New York, 2004) pp. 131-168, ISBN:1-59033-923-1

[55] V.G. Bagrov, D.M. Gitman, A. Levin, and V.B. Tlyachev, Aharonov-Bohm Eﬀect in cyclotron and synchrotron radiations, Publicação IFUSP 1395/2000; hep-th/0001108; quant-ph/001022, Nucl. Phys. B605 (2001) 425-454

[56] V.G. Bagrov, D.M. Gitman, A. Levin, and V.B. Tlyachev, Impact of Aharonov-Bohm Solenoid on Particle Radiation in Magnetic Field, Mod.Phys.Lett. A16, No. 18 (2001) 1171-1179

21 [57] V.G. Bagrov, D.M. Gitman, and V.B. Tlyachev, l-dependence of particle radiation in magnetic- solenoid ﬁeld and Aharonov-Bohm Eﬀect, Int. Journ. Mod. Phys. A17 (2002) 1045-1048

[58] V.G. Bagrov, D.M. Gitman, and V.B. Tlyachev, Aharonov-Bohm Eﬀect in Synchrotron Radiation, Proceedings of FORA, No. 5 (2000) 7-26

[59] V.G. Bagrov, V.G. Bulenok, D.M. Gitman, V.B. Tlyachev, J.A. Jara, and A.T. Jarovoi, Angular behavior of synchrotron radiation harmonics, Phys. Rev. E 69, 046502 (2004)

[60] V.G. Bagrov, V.G. Bulenok, D.M. Gitman, V.B. Tlyachev, and A.T. Jarovoi, New results in classical theory of synchrotron radiation, Surface. Roentgen, synchrotron, and neutron studies. No. 11 (2003) pp. 59-65.

[61] V.G. Bagrov, D.M. Gitman, V.B. Tlyachev, and A.T. Jarovoi, Evolution of angular distribution of polarization components of synchrotron radiation under changes of particle energy, ”Recent Problems in Field Theory”, Proceedings of XV International Summer School ”Volga ” 22.June-03. July 2003, Kazan, Russia, (Kazan, 2004) pp. 9-23

[62] V.G. Bagrov, D.M. Gitman, V.B. Tlyachev, and A.T. Jarovoi, Evolution of angular distribution of polarization components for synchrotron radiation under changes of particle energy, Proceedings of the Eleventh Lomonosov Conference on Elementary Particle Physics Particle Physics in Laboratory, Space and Universe, 21-27 August 2003, Moscow, Russia (World Scientiﬁc, New Jersey, Singapore 2005)

[63] V.G. Bagrov, D.M. Gitman, V.B. Tlyachev, A.T. Jarovoi, New theoretical Results in Synchrotron Radiation, Nuclear Instruments & Methods in Physics Research, B240 (2005) pp 638 - 645

[64] B.L. Voronov, D.M. Gitman, and I.V. Tyutin, Constructing quantum observables and self-adjoint extensions of symmetric operators I, Russian Phys. Journ. No. 1 (2007) 1-31

[65] B.L. Voronov, D.M. Gitman, and I.V. Tyutin, Constructing Quantum Observables and Self-Adjoint Extensions of Symmetric Operators II. Diﬀerential Operators, Russian Physics Journ. 50 No. 9 (2007) 853-884

[66] B.L. Voronov, D.M. Gitman, and I.V. Tyutin, Constructing quantum observables and self-adjoint extensions of symmetric operators III. Self-adjoint boundary conditions, Izv. Vuzov Fizika, 51, No. 2 (2008) 3-43 (in Russian); Russian Phys. Journ. 51, No. 2 (2008) 115-157 (English translation)

[67] B.L. Voronov, D.M. Gitman, and I.V. Tyutin, The Dirac Hamiltonian with a superstrong Coulomb ﬁeld, Theoretical and Mathematical Physics, 150(1) (2007) 34-72

[68] D.M. Gitman, A. Smirnov, I.V. Tyutin, and B.L. Voronov, Self-adjoint Schrödinger and Dirac operators with Aharonov-Bohm and magnetic-solenoid ﬁelds, arXiv:0911.0946v1 [quant-ph]

[69] D.M. Gitman, I.V. Tyutin, and B.L. Voronov, Self-adjoint extensions and spectral analysis in Calogero problem, J. Phys. A43 (2010) 145205 (34pp)

[70] D.M. Gitman, I.V. Tyutin, and B.L. Voronov, Oscillator representations for self-adjoint Calogero Hamiltonians, arXiv:0907.1736v1 [quant-ph]

10.4 Path integrals; group theory in relativistic quantum mechanics and field theory; semiclassical methods and coherent states The path integral method was used in QED with unstable vacuum to calculate the density matrix • of particles created by external fields [1, 2]. For the first time, it was constructed a path integral presenting the generating functional with two sources, which always appears in the quantum theory of fields with unstable vacuum [3, 4, 5].

A generalization for the known path integral method in perturbation theory for the case of Grass- • mannian degrees of freedom (a super-generalization) was elaborated in details and extended to the case of constrained theories [6, 7]. 22 Different kinds of propagator representations for relativistic particles were constructed through • path integrals [25, 26, 30, 31, 2, 30, 14, 34, 13, 36]. Two super-generalizations of the Schwinger proper time method were introduced for the first time. For the first time, it was shown that all Grassmann integrations can be performed in the path integral representations for spin particle propagators, as well as were obtained expressions for the so called spinor factor in arbitrary external fields [27, 30, 45]. Using the obtained path integral representations, the particle propagators were calculated in various configurations of external fields [30, 31, 4, 44, 45]. An application of the Grassmann path integrals to the calculus of operators was developed in the • work [35]. In the articles [14, 15, 16, 17, 18, 19, 20], coherent states of lie groups and its applications were • studied. For example, it was given a construction for all groups SU(N) and SU(l + 1).

References

[1] I.L. Buchbinder, D.M. Gitman and V.P. Frolov, Density matrix for particle-creation processes in external field, Izw. VUZov Fizika (Sov. Phys. Journ. 529-533) 6 (1980) 77-81. [2] V.P. Barashev, E.S. Fradkin, D.M. Gitman and Sh.M. Shvartsman, The problems of QED with unstable vacuum. Reduction formulas. The density matrix of particles creating in an external field, Trudu PhIAN (Proceedings of Lebedev Institute, Moscow) 201 (1990) 74-94. [3] I.L. Buchbinder, E.S. Fradkin and D.M. Gitman, Generating functional in quantum field theory with unstable vacuum, Preprint PhIAN (Lebedev Institute), 138 (1981). [4] D.M. Gitman and V.A. Kuchin, Generating functional of mean field in quantum electrodynamics with unstable vacuum, Izw. VUZov Fizika (Sov. Phys. Journ.) 10 (1981) 80-84.

[5] S.P. Gavrilov, D.M. Gitman and E.S. Fradkin, Quantum electrodynamics at finite temperature in presence of an external field, violating the vacuum stability, Sov. Journal Nucl. Phys. (Yadernaja Fizika), 46 (1987) 172-180. [6] D.M. Gitman and I.V. Tyutin, Canonical quantization of fields with constraints (Nauka, Moscow, 1986) pp. 1-216. [7] D.M. Gitman and I.V. Tyutin, Quantization of Fields with Constraints pp. 1—291 (Springer- Verlag, Berlin Heidelberg New-York London Paris Hong-Kong Barcelona, 1990). [8] V.G. Bagrov, D.M. Gitman and I.L. Buchbinder, Coherent states of relativistic particles, Izw. VUZov Fizika (Sov. Phys. Journ.) 8 (1975) 134-135. [9] V.G. Bagrov, D.M. Gitman and V.A. Kuchin, Eigenfunctions of linear combinations of creation and annihilation operators, Izw. VUZov Fizika (Sov. Phys. Journ.) 9 (1975) 13-19. [10] V.G. Bagrov, D.M. Gitman and V.A. Kuchin, External field in quantum electrodynamics and coherent states In Actual problems of theoretical physics (Moscow State University, Moscow, 1976) pp. 334-342. [11] V.G. Bagrov, I.L. Buchbinder and D.M. Gitman, Coherent states of a relativistic particle in an external electromagnetic field, Journ. Phys. A 9 (1976) 1955-1965. [12] V.G. Bagrov, I.L. Buchbinder, D.M. Gitman and P.M. Lavrov, Coherent states of the electron in quantized electromagnetic wave, Theor. Mat. Fiz. 33 (1977) 419-426. [13] V.G. Bagrov, I.L. Buchbinder and D.M. Gitman, Construction of coherent states for relativistic particles in external fields, In Group Theoretical Methods in Physics Proceedings of First Zvenigorod Seminar, (Nauka, Moskva, 1980) pp. 232-239. [14] D.M. Gitman and A.L. Shelepin, Coherent states related to groups SU(N) and SU(N, 1), Izw. VUZov Fizika (Sov. Phys. Journ.) 1 (1990) 83—89. 23 [15] D.M. Gitman, S.M. Carchev and A.L. Shelepin, Coherent states for groups SU(N) and SU(N, 1) and its applications in relativistic quantum theory, Trudy PhIAN (Proceedings of Lebedev Institute, Moscow), 201 (1990) 95-138.

[16] D.M. Gitman and A.L. Shelepin, Coherent states for variables angular momentum-angle, Kratk. Soob. Fiz. (Lebedev Inst.) 1 (1990) 31-33.

[17] D.M. Gitman and A.L. Shelepin, Coherent states of the SU(N) and SU(N, 1) groups and quantization on the corresponding homogeneous spaces, Preprint MIT, CTP # 1990 (1991) 1-32.

[18] D.M. Gitman and A.L. Shelepin, Coherent States of the SU(N) and SU(N, 1) groups, in Proc. XVIII International Colloquium on Group Theoretical Methods in Physics, Moscow, June 1990 (Springer-Verlag, 1990).

[19] D.M. Gitman and A.L. Shelepin, Coherent States of the SU(N) groups, Journ. Phys. A 26 (1993) 313-327.

[20] D.M.Gitman, A.L.Shelepin, Coherent States of SU(l,1) Groups, Journ. Phys. A 26 (1993) 7003- 7018.

[21] J.P. Gazeau, M.C. Baldiotti, and D.M. Gitman, Coherent states of a particle in magnetic ﬁeld and Stieltjes moment problem, Physics Letters A 373 (2009) 1916-1920

[22] D.M. Gitman and A.L. Shelepin, Representations of SU(N) groups on the polynomials of the an- ticommuting variables, Kratk. Soob. Fiz. (Lebedev Institute) No.11 (1998) pp.21-30

[23] V.G. Bagrov, D.M. Gitman and V.D. Skarginski, Aharonov-Bohm effect for quantum states of relativistic electron in homogeneous magnetic field and in thin solenoid field, Preprint PhIAN (Lebedev Institute), 101 (1986) 1-18.

[24] V.G. Bagrov, D.M. Gitman and V.D. Skarginski, Aharonov-Bohm eﬀect for stationary and coherent states of an electron in homogeneous magnetic ﬁeld, Trudy PhIAN (Proceedings of Lebedev Institute, Moscow), 176 (1986) 151-166.

[25] E.S. Fradkin, D.M. Gitman and Sh.M. Shvartsman, Path integral for relativistic particle theory, Europhys. Lett. 15 (3) (1991) 241-244.

[26] E.S. Fradkin and D.M. Gitman Path integral representation for the relativistic particle propagators and BFV quantization, Phys. Rev. D 44 (1991) 3230-3236.

[27] D.M. Gitman, Sh.M.Shvartsman, Spinor and isospinor structure of relativistic particle propagator, Preprint IC/93/197, pp.1-8; hep-th/9310142; Phys. Lett. B 318 (1993) 122-126; Errata, Phys. Lett. B 331 (1994) 449,450

[28] D.M. Gitman, Sh.M.Shvartsman and W.da Cruz, Path Integral over Velocities for Relativistic Particle Propagator, Bras. Journ. Phys. 24, No.4 (1994) 844-854.

[29] D.M. Gitman, S.I. Zlatev and W.da Cruz, Spin Factor and Spinor Structure of Dirac Propagator in Constant Field, Bras. Journ. Phys. 26 (1996) 419-425

[30] D.M. Gitman, Pseudoclassical Theory of Relativistic Spinning Particle, Preprint IFUSP/P-1173, pp.1-27, September/1995, in Topics in Statistical and Theoretical Physics, F.A. Berezin Memorial vol. American Mathematical Society Translation, Series 2, vol. 177, pp. 83-104, Amer. Math. Soc., Providence, RI, 1996

[31] D.M. Gitman and S.I. Zlatev, Spin factor in path integral representation for Dirac propagator in external ﬁeld, Phys. Rev. D55 (1997) 7701-7714

[32] S.P. Gavrilov and D.M. Gitman, Proper time and path integral representations for the commutation function, J. Math. Phys. 37 (7) (1996) 3118-3130

24 [33] D.M. Gitman, Path integrals and pseudoclassical description for spinning particles in arbitrary dimensions, Nucl. Phys. B 488 (1997) 490-512

[34] D.M. Gitman, Path integrals and pseudoclassical description for spinning particles in arbitrary dimensions, in Functional Integration: Basics and Applications, Ed. C. DeWitt-Morette, P. Cartier, and A. Folacci, NATO ASI Series B: Physics Vol.361, p 418, (Plenum Publishing Corp. 1997)

[35] D.M. Gitman, S.I. Zlatev and P.B. Barros, Application of Path Integration to Operator Calculus, J. Phys. A: Math.Gen. 31 (1998) 7791-7799

[36] D.M. Gitman and S.I. Zlatev, Semiclassical Form of the Relativistic Particle Propagator, Mod. Phys. Lett.A 12 (1997) 2435-2443

[37] V.G. Bagrov, S.P. Gavrilov, D.M. Gitman, and D. P. Meira Filho, Coherent states of spinless particle in large magnetic-solenoid ﬁeld, Problems of Modern Theoretical Physics (A volume in honour of Professor I.L. Buchbinder in the occasion of his 60th birthday) (Tomsk State University Press, Toms 2008) pp 57-77; ISBN 978-5-89428-280-0

[38] V.G. Bagrov, S.P. Gavrilov, D.M. Gitman, and D. P. Meira Filho, Coherent states of non-relativistic electron in magnetic—solenoid ﬁeld, e-Print: arXiv:1002.2256 [quant-ph], submitted to Journ. Physics. A

[39] V.G. Bagrov, S.P. Gavrilov, D.M. Gitman, and D. P. Meira Filho, Coherent states of relativistic electron in magnetic—solenoid ﬁeld, to be published

10.5 Classical and pseudoclassical relativistic particle models and their quantization It was presented a generalization for the pseudoclassical action of a spin particle in the presence • of an anomalous magnetic moment [1, 2].

New pseudoclassical models for relativistic particles were proposed: for a Weyl particle [3]; for • spin particles in 2+1 dimensions [6, 25, 26]; for the Weyl particle in 10 dimensions [7, 8, 18, 19]; for spin 1 massive particles (Chern-Simons particles) [9] and for massive particles with higher spins (integer and half-integer) in 2+1 dimensions [10, 11] (both supersymmetric models); for Dirac Massive particles (spin 1/2) in arbitrary odd dimensions [12, 13].

It was considered a consistent canonical quantization procedure for pseudoclassical models of • relativistic spin 1 particles. The constructed quantum mechanics for the massive case proves to be equivalent to the Proca theory and, for the massless case, to the Maxwell theory [5].

The propagator for a spinorial particle in external abelian ﬁelds in arbitrary dimensions was • presented by means of path integral [14].

It was constructed a path integral representation for the commutation function of the quantized • spinorial ﬁeld interacting with arbitrary external electromagnetic ﬁelds [46].

For the ﬁrst time, the canonical quantization of spin 0, 1/2 and 1 relativistic particles was per- • formed through a new proposed temporal gauge[15, 16, 17], and for the ﬁrst time a consistent relativistic quantum mechanics was constructed.

Other relevant works in this area are [27, 28, 29]. • References

[1] D.M.Gitman, A.V.Saa, Pseudoclassical Model of Spinning Particle with Anomalous Magnetic Mo- mentum, Mod. Phys. Lett. A, 8 (1993) 463-468.

[2] D.M.Gitman, A.V.Saa, Quantization of Spinning Particle with Anomalous Magnetic Momentum, Class. Quantum Grav. 10 (1993) 1447-1460. 25 [3] D.M. Gitman, A.E. Gonçalves and I.V. Tyutin, New pseudoclassical model for Weyl particles, Phys. Rev. D 50 (1994) 5439-5442.

[4] D.M. Gitman, Sh.M.Shvartsman and W.da Cruz, Path Integral over Velocities for Relativistic Par- ticle Propagator, Bras. Journ. Phys. 24, No.4 (1994) 844-854.

[5] D.M. Gitman, A.E. Gonçalves and I.V. Tyutin, Quantization of a pseudoclassical model of the spin 1 relativistic particle, Int.J.Mod.Phys. A 10 (1995) 701-718.

[6] D.M. Gitman, A.E. Gonçalves and I.V. Tyutin, Pseudoclassical Supergauge Model for 2+1 Dirac Particle, Physics of Atomic Nuclei, 60 No.4 (1997) 748-752

[7] D.M. Gitman and A.E. Gonçalves, Pseudoclassical model for Weyl particle in 10 dimensions, J. Math. Phys. 38 (5) (1997) 2167-2170

[8] D.M. Gitman, Pseudoclassical Theory of Relativistic Spinning Particle, in Topics in Statistical and Theoretical Physics, F.A. Berezin Memorial vol. American Mathematical Society Translation, Series 2, vol. 177, pp. 83-104, Amer. Math. Soc., Providence, RI, 1996

[9] D.M. Gitman and Tyutin, Pseudoclassical model for Chern-Simons particles, Mod. Phys. Lett. A11 (1996) 381-388

[10] D.M. Gitman and Tyutin, Pseudoclassical description of higher spins in 2+1 dimensions, Int. J.Mod.Phys.A 12 (1997) 535-556

[11] D.M. Gitman and I.V. Tyutin, Pseudoclassical description of higher spins in 2+1 dimensions, in Proceedings of SECOND INTERNATIONAL SAKHAROV CONFERENCE ON PHYSICS, Moscow, Russia, 20-24 May 1996, ed. I.M. Dremin, A.M. Semikhatov, (World Sci. Singapore, 1997) 428-434

[12] D.M. Gitman and A.E. Gonçalves, Pseudoclassical description of the massive Dirac particles in odd dimensions, Int. J. Theor.Phys. 35 (1996) 2427-2438

[13] D.M. Gitman, Quantization of Spinning Particles in Odd Dimensions, Nucl. Phys. B (Proc. Suppl.) 57 (1997) 231-234

[14] D.M. Gitman, Path integrals and pseudoclassical description for spinning particles in arbitrary dimensions, Nucl. Phys. B 488 (1997) 490-512

[15] D.M. Gitman and I.V. Tyutin, Canonical quantization of the relativistic particle, JETP Lett. 51, v. 4 (1990) 214; Pis’ma Zh. Eksp. Teor. Fiz. 51, v. 3 (1990) 188—190.

[16] D.M. Gitman and I.V. Tyutin, Classical and quantum mechanics of the relativistic particle, Class. and Quantum Grav. 7 (1990) 2131-2144.

[17] G. Fulop, D.M. Gitman and I.V. Tyutin, Reparametrization Invariance as Gauge Symmetry, Preprint IFUSP/P-1263, pp.1-30, April/1997; hep-th/9805040; Int. J. Theor. Phys. 38 (1999) 1953- 1980

[18] D.M. Gitman, A.E. Gonçalves and I.V. Tyutin, Remark to the Comment on “New pseudoclassical model for Weyl particles, Preprint FIAN/TD/96-03; hep-th/9602151

[19] D. M. Gitman, and I.V. Tyutin, A pseudo-classical model of a Weyl particles and quantization of classical constants, Russian Physics Journal 45, No.7 (2002) pp. 690-694

[20] A.A. Deriglazov and D.M. Gitman, Classical description of spinning degrees of freedom of relativistic particles by means of commuting spinors, Publicação IFUSP 1324/98; hep-th/9811229; Mod. Phys. Lett. A14 (1999) pp. 709-720

[21] S.P. Gavrilov, D.M. Gitman, Quantization of Point-Like Particles and Consistent Relativistic Quantum Mechanics, Int. J. Mod. Phys. A15 (2000) 4499-4538

26 [22] S.P. Gavrilov, D.M. Gitman, Quantization of the Relativistic Particle, Class.Quant.Grav. 17 issue 19 (2000) L133-L139 [23] S.P. Gavrilov, D.M. Gitman, Quantization of the Relativistic Particle and Consistent Relativistic Quantum Mechanics, Proceedings of the International Conference ”Quantization, gauge theories, and strings, Moscow, Russia, June 5-10, 2000” dedicated to the memory of Professor Eﬁm Fradkin, Ed. A. Semikhatov, M. Vasiliev, V. Zaikin, v.II (Scientiﬁc World, 2001) pp.27-35. [24] S.P. Gavrilov, D.M. Gitman, Quantization of a spinning particle in an arbitrary background, Class.Quant.Grav. 18 (2001) 2989-2998 [25] R. Fresneda, S. Gavrilov, D. Gitman, and P. Moshin, Quantization of ( 2 + 1)-spinning particles and bifermionic constraint problem, Class.Quant.Grav.21 (2004) pp.1419-1442 [26] S.P. Gavrilov, D.M. Gitman, and J.L. Tomazelli, Comments on spin operators and spin-polarization states of 2 + 1 fermions, Eur. Phys. J. C (2005) DOI: 10.1140/epjc/s2004-02026-9 [27] D.M. Gitman, Berezin-Marinov’s pseudoclassical action,”Reminiscences about Felix Berezin- founder of supermathematics”, ed. by E. Karpel, P.A. Minlos, I.V. Tyutin, and D.A. Leites, and (M(TZ)NMO, Moscow 2009) pp. 139-148; ISBN 978-5-94057-458-3 [28] R. Fresneda and D. Gitman, Pseudoclassical description of scalar particle in non-Abelian background and path-integral representations, Intern. Journ. Mod. Phys. A 23 (6) (2008) 835-853. [29] D. Gitman and V.G. Kupriyanov, Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin-Marinov action, Europ. Phys. J. C 54 (2008) 325-332

10.6 Theory of higher spins A general approach to higher spins description was made in which a scalar field under the Poincaré • group is considered as a generating function for conventional multicomponent fields. This approach allows an unified consideration to the problem of construction of different kinds of relativistic wave equations and justifies the use of methods from group theory [1, 2, 3, 10, 13, 14, 15]. It was proposed a quantum-mechanical description of relativistic orientable objects. It generalizes • Wigner’s ideas in relation to the treatment of non-relativistic orientable objects (in particular, a non-relativistic rotator) using two reference frames (one fixed in space and the other fixed in the body) [16, 17]. Other relevant works: [3, 4, 5, 6, 7, 8, 9, 11, 12]. • References

[1] D.M.Gitman, A.L.Shelepin, 2+1 Poincare group and relativistic wave equations, Proceedings of VII International Conference on Symmetry Methods in Physics, Dubna, July 1995, v1, pp.212-219, (JINR, Dubna 1996). [2] D.M.Gitman, A.L.Shelepin, Poincare group and relativistic wave equations in 2+1 dimensions, J. Phys. A: Math. Gen. 30 (1997) 6093-6121 [3] D.M. Gitman and I.V. Tyutin, Pseudoclassical description of higher spins in 2+1 dimensions, in Proceedings of SECOND INTERNATIONAL SAKHAROV CONFERENCE ON PHYSICS, Moscow, Russia, 20-24 May 1996, ed. I.M. Dremin, A.M. Semikhatov, (World Sci. Singapore, 1997) 428-434 [4] A.V. Galajinsky and D.M. Gitman, Siegel superparticle, higher order fermionic constraints, and path integrals, hep-th/9805044, Preprint IFUSP/P-1308, pp.1-22, Maio/1998; Nucl. Phys. B536 (1999) 435-453 [5] A.A. Deriglazov and D.M. Gitman, The Green-Schwarz type formulation of D=11 S-invariant su- perstring and superparticle action, Preprint IFUSP/P-1304, pp.1-30, Abril/1998; hep-th/9804055, Int. J. Mod. Phys. A14, No.17 (1999) 2769-2790 27 [6] A.A. Deriglazov, Galajinsky, and D.M. Gitman, On zero modes of the eleven dimensional su- perstring, Preprint IFUSP/P-1298, pp.1-7, Março/1998; hep-th/9801176; Phys. Rev. D59 (1999) 048902(4)

[7] A.A. Deriglazov, A.V. Galajinsky, and D.M. Gitman, Massless chiral multiplet model as ﬁrst quantized AB—superparticle, Proceedings of Second International Conference ”Quantum Field theory and Gravity” (July 28—August 2, 1997, Tomsk), Tomsk, Russian Federation 1998, pp. 164—172, Eds. I. Buchbinder and K. Osetrin

[8] A.A. Deriglazov and D.M. Gitman, Examples of D=11 S-supersymmetric Actions for Point-Like Dynamical Systems, Mod. Phys. Lett. A13 (1998) 2559-2570

[9] A.V. Galajinsky and D.M. Gitman, On minimal coupling of the ABC-superparticle to supergravity background, Phys. Rev. D59 (1999) 047504

[10] D.M. Gitman, and A. Shelepin, Fields on the Poincaré Group: Arbitrary spin description and relativistic wave equations, Int.J.Theor.Phys. 40 (2001) 603-684

[11] I.L. Buchbinder, D.M. Gitman, V.A. Krykhtin, and V.D. Pershin, Equations of motion for massive spin 2 ﬁeld coupled to gravity, Nucl.Phys. B584 No.1-2 (2000) 615-640

[12] I.L. Buchbinder, D.M. Gitman, and V.D. Pershin, Causality of Massive Spin 2 Field in External Gravity,Phys.Lett.B492 (2000) 161-170

[13] D.M. Gitman, and A. Shelepin, Z-description of the relativistic spin, Proceedings of XXIII Interna- tional Colloquium on Group Theoretical Methods in Physics, Edited by A.N.Sissakian, G.S.Pogosyan and L.G.Mardoyan, V.2 (Dubna, JINR, 2002) pp.376-384

[14] I.L. Buchbinder, D.M. Gitman, and A.L. Shelepin, Discrete symmetry transformations as auto- morphisms of the proper Poincare group, Int. J. Theor. Phys. 41, No. 4 (2002) 753-790

[15] D.M. Gitman, and A. Shelepin, Z-description of the relativistic spin, Hadronic Physics, No. 3,4 (Special Issue on HIGHER SPINS, QCD, AND BEYOND) (2003) pp.259-274

[16] D.M. Gitman and A. Shelepin, Field on Poincaré Group and Quantum Description of Orientable Objects, Europ. Physical Journal C, 61, Issue1 (2009)111

[17] D.M. Gitman and A.L. Shelepin, Classiﬁcation of quantum relativistic orientable objects, arXiv:1001.5290v1 [hep-th], submitted to Class. Quantum Grav.

10.7 Theory of two and four levels systems and applications to the quantum computation It was presented a detailed study of the spin equation , that is, the two levels system described • by two time dependent coupled diﬀerential equations. 26 new classes of exact solutions for those systems were obtained [1, 3, 4, 5].

A systematic method for the obtention of new solutions of the spin equation starting from a previ- • ously known solution was developed using an adaptation of the Darboux transformation method for the differential equation that describes a two levels system. The existence of transformations under which the form of the equations of the two levels systems is invariant was demonstrated. In particular, a Darboux operator that transforms a problem given by real fields int a new problem also given by real fields was constructed. [2, 3].

It was developed a detailed study of the equation for two coupled spins (four levels systems) • and, specially, it was demonstrated how it is possible to construct exact solutions of this problem starting from known solutions of the two levels system [6].

It was developed a method for the obtention of exact solutions of the spin equation for the • important case of external fields whose effective influence is restrict to a finite time interval [8].

28 Using the exact two and four levels systems, it was described the theoretical implementation of a • set of quantum universal logical gates, studying specially the characteristics of the external ﬁelds and the possible practical scenarios for the implementations of such devices [7, 9].

References

[1] V.G. Bagrov, J.C.A. Barata, D.M. Gitman, and W.F. Wreszinski, Aspects of Two-Level Systems under External Time Dependent Fields , J. Phys. A34 (2001) 10869-10879

[2] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, and V.V. Shamshutdinova, Darboux transformation for two-level system, Annalen der Physik 14 (2005) 390-397

[3] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, and A.D. Levin, Spin equation and its solutions, An- nalen der Physik 14 [11-12] (2005) pp.764-789

[4] D.M. Gitman, B.F. Samsonov, V.V. Shamshutdinova, Polynomial pseudo-supersymmetry underlying a two-level atom in an external electromagnetic ﬁeld, Czech. J. Phys. 55, No.9 (2005) pp. 1173-1176

[5] B.F. Samsonov, V.V. Shamshutdinova, D.M. Gitman, Two-level systems: exact solutions and underlying pseudo-supersymmetry, Ann. Phys. N.Y. 322 (2007) 1043-1061

[6] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, and A.D. Levin, Two interacting spins in external ﬁeld. Four-level systems, Annalen der Physik, 16, 8 (2007) 274-285

[7] M.C. Baldiotti, D.M. Gitman, Four-level systems and a universal quantum gate, Annalen der Physik (Berlin) 17, pp. 450-459 (2008)

[8] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, and A.D. Levin, Two and four-level systems in magnetic ﬁelds restricted in time, Publicação IFUSP ; e-Print: arXiv:0803.0299, submitted to Annalen der Physik

[9] M.C. Baldiotti, V.G. Bagrov, and D.M. Gitman, Two Interacting Spins in External Fields and Application to Quantum Computation, Physics of Particles and Nuclei Letters, 6, No. 7 (2009) pp. 559-562

10.8 Quantum mechanics and field theory in non-commutative spaces A Moyal plane whose space non-commutativity parameter θν is constructed by two bifermionic • parameters (Grassmann algebra elements) was introduced. In this approach, the Moyal product contains a finite number of derivatives, what permits to avoid difficulties in relation to the standard procedure. The renormalizability properties of non-commutative theories of this kind were analyzed [1, 3].

The construction of (pseudo) classical θ-modiﬁed actions (modiﬁed by the coordinate non-commutativity) • for the relativistic scalar and spinorial particles was discussed. The classical and quantum dynamics of a charged particle in non-commutative spaces was considered [2, 4, 5].

It was analyzed the modification of the energy levels of the relativistic hydrogen atom due to • the space non-commutativity. The θ-modification of the Pauli equation was constructed. The θ-modified interaction between non-relativistic spin and the magnetic field was constructed along with a θ-modification of the Heisenberg model for coupled spins [6, 7].

References

[1] D. M. Gitman, D. V. Vassilevich, Space-time noncommutativity with a bifermionic parameter, Mod. Phys. Lett. 23 (12) (2008) 887-893

[2] D. Gitman and V.G. Kupriyanov, Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin-Marinov action, Europ. Phys. J. C 54 (2008) 325-332 29 [3] R. Fresneda, D.M. Gitman, and D.V. Vassilevich, Nilpotent noncommutativity and renormalization, Phys. Rev. D78: 025004 (2008)

[4] D.M. Gitman and V.G. Kupriyanov, Gauge Invariance and Classical Dynamics of Noncommutative Particle Theory, Journal of Mathematical Physics, 51, 022905 (2010) 022905 (1-8)

[5] J.P. Gazeau, M.C. Baldiotti, and D.M. Gitman, Semiclassical and quantum motion on non- commutative plane, Physics Letters A, DOI information: 10.1016/j.physleta.2009.08.059

[6] T.C. Adorno, M.C. Baldiotti, M. Chaichian, D.M. Gitman, and A. Tureanu, Dirac Equation in Noncommutative Space for Hydrogen Atom, Phys. Letters B 682, Issue 2, (2009) 235-239

[7] T.C. Adorno, M.C. Baldiotti, and D.M. Gitman, Quantum and pseudoclassical description of non- relativistic spinning particles in noncommutative space, submitted to Physics Lett. B

10.9 Quantum Statistics from 1966 to 1970, he worked in the ﬁeld of Quantum Statistics, solving the following problems:

New kinds of integral-diﬀerential equations where proposed to distribution functions in classical • and quantum statistics[1, 3, 5]

A new form for the Wigner-like distribution functions was proposed to a quantum system in • thermal equilibrium. Based in this representation, corrections were calculated to the classical distribution.[2].

Variational principles for quantum statistics were studied in the articles [4, 5, 6, 7, 8]. In particular, • a variational principle for the thermodynamic potential was constructed. In fact, this was one of the ﬁrst works in which the idea of eﬀective action was introduced and in which, in particular, was demonstrated in detail the case of the quantum statistics.

References

[1] E.A. Arinshtein and D.M. Gitman, System of integral equations for partial distribution functions, Izw. VUZov Fizika (Sov. Phys. Journ.) 9 (1967) 110-113.

[2] E.A. Arinshtein and D.M. Gitman, Temperature dependence of quantum distribution functions, Izw. VUZov Fizika (Sov. Phys. Journ.) 9 (1967) 113-120.

[3] E.A. Arinshtein and D.M. Gitman, Integral equations for partial density matrices, Izw. VUZov Fizika (Sov. Phys. Journ.) 8 (1968) 81-86.

[4] E.A. Arinshtein and D.M. Gitman, A variational principle for mean occupation numbers, Izw. VUZov Fizika (Sov. Phys. Journ.) 10 (1968) 146-147.

[5] D.M. Gitman, A system of integral equations for partial density matrices, Izw. VUZov Fizika (Sov. Phys. Journ.) 12 (1969) 155-158.

[6] D.M. Gitman, An expression for the thermodynamical potential in the form of a stationary functional on mean occupation numbers, Izw. VUZov Fizika (Sov. Phys. Journ.) 4 (1970) 96-102.

[7] D.M. Gitman and A.G. Tchernishov, A variational principle for the thermodynamical potential of a two-component system Izw. VUZov Fizika (Sov. Phys. Journ.) 3 (1971) 30-35

[8] E.A. Arinshtein and D.M. Gitman, Equations for generating functional in classical and quantum mechanics, Izw. VUZov Fizika (Sov. Phys. Journ.) 9 (1971) 98-102.

30 10.10 Other subjects References

[1] D. Gitman, My Encounters with Felix Alexandrovich Berezin: Snapshots of Our Life in the 1960s, ’70s and Beyond, in ”FELIX BEREZIN. Life and Death of the Mastermind of Supermath- ematics”, ed. by M. Shifman (World Scientiﬁc, Singapore 2007) pp 181-205; russian translation in ”Reminiscences about Felix Berezin-founder of supermathematics”, ed. by E. Karpel, P.A. Minlos, I.V. Tyutin, and D.A. Leites, and (M(TZ)NMO, Moscow 2009) pp. 282-301

[2] D.M. Gitman, I.V. Tyutin,J.L. Assirati, and M.G. da Costa, Structure of Lorentz transformation of general form, Gravitation and Cosmology 4 No.2(14) (1998) 163-166

[3] E.R. Berdichevkaja, S.P. Gavrilov and D.M. Gitman, A mathematical model for calculation of the traﬃc capacity of machinery for production of integral schemes, Elektronaja Technika, 7 1 (1979) 83-90.

[4] D.M. Gitman, Quantization, pp. 311-312; Constraint, General, p. 112; Dirac Quantization Rules, pp. 129-130; Constraint Gauge Theories, pp. 109-112, ”CONCISE ENCYCLOPEDIA OF SU- PERSYMMETRY”, Eds. S. Duplij, W.Siegel, J.Bagger, (Kluwer Acad. Publisher, Dordrecht Boston London, 2003)