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 1110.1 Quantum field theory with external backgrounds . . . . . . . . . . . . . . . . . . . . . . 1110.2 General theory of constrained systems and their quantization . . . . . . . . . . . . . . . 1510.3 Exact solutions of the relativistic wave equations and theory of self-adjoint extensions . 1810.4 Path integrals; group theory in relativistic quantum mechanics and field theory; semi-

classical methods and coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.5 Classical and pseudoclassical relativistic particle models and their quantization . . . . . 2510.6 Theory of higher spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.7 Theory of two and four levels systems and applications to the quantum computation . . 2810.8 Quantum mechanics and field theory in non-commutative spaces . . . . . . . . . . . . . 2910.9 Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.10 Other subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1 Personal information

Born on July 2, 1944, Tashkent, Uzbekistan - USSRBrazilian CitizenshipLanguages: Russian, English, Portuguese and GermanAddress: Instituto de Física da USP, Departamento de Física Nuclear, Instituto de Física, Universidadede São Paulo, C.P. 66318, CEP 05315-970, São Paulo, SP, BrasilTelephone: 55-11-3091-6948.E-mail: gitman@dfn.if.usp.br, dmitrygitman@hotmail.comHomepage: http://www.dfn.if.usp.br/pesq/tqr/pt/gitman

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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 forDistribution 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 QuantumStatistics” — Tomsk State University, Russia. Diploma MFM No. 011435, State Higher Attesta-tion Commission, Moscow, January 16, 1970. Received official favorable opinions from ProfessorsV.L. Pokrovsky - Landau Institute, Chernogolovka: M. Gaigasian — Tomsk State University andProf. A.A. Sokolov, Moscow State University. This title was recognized by University of SãoPaulo 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, April1, 1976.

• 1979: Received the title “Doctor in Sciences”, (the highest scientific degree in Physics andMathematics given in Russia) with the thesis “Problems of External Fields in QuantumElectrodynamics”, Institute of Nuclear Physics of Novosibirsk. Diploma FM No. 001066,“Higher Attestation Commission”, Moscow, April 18, 1980. Received official favorable opin-ions from Professors F.A. Berezin, Moscow State University, B.N. Baier, Institute of NuclearPhysics, Novosibirsk, V.N. Barbashov, International Institute of Nuclear Research, Dubna, andV.I. Manko, Lebedev Physical Institute, Moscow. This title was recognized by University ofSão 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 TheoreticalPhysics”. 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 AutomationControl Systems and Radio Engineering (TIASUR), Tomsk, Russia.

• 1970-1975: Associate Professor at the Department of Physics — Tomsk Institute of AutomationControl 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, PhysicsInstitute — 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.

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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, forgraduated 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 particlequantization, (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)

1TIASUR-Tomsk Institute of Automation Control Systems and Radio Engineering; TSPU-Tomsk State PedagogicalUniverstity; MIREA-Moscow Institute of Radio Engineering, Electronics and Automation; TSU-Tomsk State University;USP-University of Sao Paulo

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Doctorates

1. P.V. Bozrikov. Thesis: Motion of an electron in the quantized electromagnetic plane wave, (TomskState 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, (TomskState University, Tomsk, 1975);

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

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

6. I.M. Lichtzier. Thesis: Green’s functions and one-loop effective action in external gauge andgravitational 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 PhysicalInstitute, Moscow, 1990).

10. Antonio Edson Gonçalves. Thesis: Pseudoclassical models and their quantizations, (University ofSão 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 modernquantum theory, (University of São Paulo, São Paulo, 1998)

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

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

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

16. Rodrigo Fresneda, Thesis: Some problems of quantization in theories with non-Abelian back-grounds 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-commutativequantum mechanics, (University of São Paulo, São Paulo, march 2009)

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

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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 supersym-metric extended objects

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

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

5. 2002-2009: Pavel Moshin, Project: Problems of Lagrangian and Hamiltonian BRST quantizationof 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, inprogress.

6 Books published and in progress

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

2. Canonical Quantization of Fields with Constraintswith I. V. TyutinNauka, Moscow (1986)216 pages (in russian)

3. Exact Solutions of Relativistic wave Equationswith V. G. BagrovKluwer Acad. Publish., Dordrecht, Boston, London (1990)321 pages

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4. Quantum Electrodynamics with Unstable Vacuumwith E. S. Fradkin e Sh. M. ShvartsmanNauka, Moscow (1991)294 pages (in russian)

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

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

7. Physical Observables in Non-trivial Quantum Systems (Constructing self-adjoint op-erators and solving spectral problems), with I. V. Tyutin and B. Voronov, to be publishedin 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 bySpringer Verlag.

7 Selected articles

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

1. D.M. Gitman and A. Shelepin, Field on Poincaré Group and Quantum Description of OrientableObjects, 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 (inparticular, 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) amountsto introducing wave functions that depend on elements of the Poincaré group G. A completeset of transformations that test the symmetries of an orientable object and of the embeddingspace belongs to the group Π = G×G. All such transformations can be studied by considering ageneralized regular representation of G in the space of scalar functions on the group, f(x, z), thatdepend on the Minkowski space points x ∈ G/Spin(3, 1) as well as on the orientation variablesgiven by the elements z of a matrix Z ∈ Spin(3, 1). In particular, the field f(x, z) is a generating

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function of usual spin-tensor multicomponent fields. In the theory under consideration, there arefour 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 inQuantum Field Theory, Phys. Rev. Lett. 101, 130403(4) (2008).QFT with an external background can be considered as a consistent model only if backreaction isrelatively small with respect to the background. To find the corresponding consistency restrictionson an external electric field and its duration in QED and QCD, we analyze the mean energy densityof quantized fields for an arbitrary constant electric field E, acting during a large but finite timeT . Using the corresponding asymptotics with respect to the dimensionless parameter eET 2, onecan see that the leading contributions to the energy are due to the creation of particles by theelectric field. Assuming that these contributions are small in comparison with the energy densityof 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 Coulombfield, 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 theCoulomb 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-calledcritical charge with Z = α−1 = 137 based on the fact that the standard expression for the lowerbound state energy yields complex values at overcritical charges. We show that from the math-ematical standpoint, there is no problem in defining a self-adjoint Hamiltonian for any value ofcharge. 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 overcriticalcharges is a non uniqueness of the self-adjoint Hamiltonian, but this non uniqueness is also char-acteristic for charge values less than the critical one (and larger than the subcritical charge withZ = (

√3/2)α−1 = 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 extensionsof symmetric operators and the Krein method of guiding functionals. The relation of the con-structed one-particle quantum mechanics to the real physics of electrons in superstrong Coulombfields 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 back-ground, Phys. Rev. D78, 045017(35) (2008).We have obtained non-perturbative one-loop expressions for the mean energy-momentum tensorand current density of Dirac’s field on a constant electric-like background. One of the goals of thiscalculation is to give a consistent description of back-reaction in such a theory. Two cases of ini-tial states are considered: the vacuum state and the thermal equilibrium state. First, we performcalculations for the vacuum initial state. In the obtained expressions, we separate the contribu-tions due to particle creation and vacuum polarization. The latter contributions are related tothe Heisenberg—Euler Lagrangian. Then, we study the case of the thermal initial state. Here, weseparate the contributions due to particle creation, vacuum polarization, and the contributionsdue to the work of the external field on the particles at the initial state. All these contributionsare 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 underwhich QED with a strong constant electric field is consistent. Under such restrictions, one canneglect the back-reaction of particles created by the electric field. Some of the obtained resultsgeneralize the calculations of Heisenberg—Euler for energy density to the case of arbitrary strongelectric 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

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structure of a theory in the Hamiltonian formulation. We demonstrate that the symmetry struc-ture 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 thehelp of a special orthogonal basis for the constraints. Thus, we succeed in describing all the gaugetransformations of a given action. We find the gauge charge as a decomposition in the orthogonalconstraint basis. Thus, we establish a relation between the constraint structure of a theory andthe structure of its gauge transformations. In particular, we demonstrate that, in the generalcase, the gauge charge cannot be constructed with the help of some complete set of first-classconstraints alone, because the charge decomposition also contains second-class constraints. Theabove-mentioned procedure of solving the symmetry equation allows us to describe the structureof an arbitrary symmetry for a general singular action. Finally, using the revealed structure of anarbitrary 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-invarianton the extremals, and the other requires that physical functions commute with FCC (the Diracconjecture).

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 thestability of vacuum (particle-creating backgrounds). Our purpose is to study the exact form ofthe final quantum state (the density operator at the final instant of time) that has emerged from agiven arbitrary initial state (from a given arbitrary density operator at the initial time instant) inthe course of evolution. We find a generating functional that allows one to obtain density operatorsfor 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 betweenthese functions and the reduced density operators. It is shown that in the general case a presenceof 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 thecomplete system is not in thermal equilibrium. In this respect, we discuss some cases whenpair-creation by an electric-like field can mimic the one-particle thermal distribution. We applyour 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 themean numbers of created particles, provided that the effects of switching the external field onand off are negligible. It is demonstrated that at high temperatures and in slowly varying electricfields 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 timederivatives in the Lagrange function (we call such coordinates degenerate). We advocate that itis reasonable to reconsider the conventional definition of singularity based on the usual Hessianand, moreover, to simplify the conventional Hamiltonization procedure. In particular, in sucha procedure, it is not necessary to complete the degenerate coordinates with the correspondingconjugate momenta.

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

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is shown that new spectral lines appear in the radiation spectrum. Due to angular distributionpeculiarities of the radiation intensity, these lines can in principle be isolated from basic cyclotronand synchrotron radiation spectra.

9. S.P. Gavrilov, D.M. Gitman, Quantization of Point-Like Particles and Consistent RelativisticQuantum 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 isrelated to a principally new realization of the Hilbert space. It allows one not only to includearbitrary backgrounds in the consideration but to get in course of the quantization a consistentrelativistic quantum mechanics, which reproduces literally the behavior of the one-particle sectorof the corresponding quantum field. In particular, in a physical sector of the Hilbert space acomplete 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 arbitrarydimensions, Nucl. Phys. B 488 (1997) 490-512.The propagator of a spinning particle in external Abelian field and in arbitrary dimensions ispresented by means of a path integral. The problem has distinct solutions in even and odd di-mensions. 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 pathintegral has a form of the standard (Berezin-Marinov) pseudoclassical action. In odd dimensionsthe solution is presented for the first time and, in particular, it turns out that the gauge invari-ant part of the effective action differs from the standard one. We propose this new action as acandidate to describe spinning particles in odd dimensions. Studying the hamiltonization of thepseudoclassical 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 considerationis 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. Inall the cases special sets of exact solutions of Dirac equation (IN- and OUT- solutions) are con-structed. Using them, characteristics of the effect are calculated. The time and dimensionalanalysis of the vacuum instability is presented. It is shown that the distributions of particlescreated by quasi-constant electric fields can be written in a form which has a thermal characterand seams to be universal, i.e. is valid for any theory with quasi-constant external fields. Its ap-plication, for example, to the particles creation in external constant gravitational field reproducesthe Hawking temperature exactly.

12. D.M. Gitman and S.I. Zlatev, Spin factor in path integral representation for Dirac propagator inexternal field, Phys. Rev. D55 (1997) 7701-7714.We study the spin factor problem both in 3 + 1 and 2 + 1 dimensions which are essentiallydifferent for spin factor construction. Doing all Grassmann integrations in the correspondingpath integral representations for Dirac propagator we get representations with spin factor inarbitrary external field. Thus, the propagator appears to be presented by means of bosonic pathintegral only. In 3 + 1 dimensions we present a simple derivation of spin factor avoiding someunnecessary 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 surprisingpossibility of complete integration over Grassmann variables gets clear. In 2 + 1 dimensions thederivation of the spin factor is completely original. Then we use the representations with spinfactor for calculations of the propagator in some configurations of external fields. Namely, inconstant 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 propaga-tors and BFV quantization, Phys. Rev. D 44 (1991) 3230-3236.

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The path-integral representations for the propagators of scalar and spinor fields in an externalelectromagnetic field are derived. The Hamiltonian form of such expressions can be interpretedin the sense of Batalin-Fradkin-Vilkovisky quantization of one-particle theory. The Lagrangianrepresentation as derived allows one to extract in a natural way the expressions for the correspond-ing gauge-invariant (reparametrization- and supergauge-invariant) actions for pointlike scalar andspinning particles. At the same time, the measure and ranges of integrations, admissible gaugeconditions, and boundary conditions can be exactly established.

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

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

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 fieldsFAPESP thematic project 1996/07134-8 (1996-2002)

• Some current problems in quantum field theoryFAPESP thematic project 2002/00222-9 (2002-2008)

• Quantization and problems in quantum field theoryFAPESP thematic project 2007/03726-1 (2008-2012)

• Problems of quantization of non-trivial classical modelsCAPES/COFECUB program, No 566/07 (2006-2009)

• Modern aspects of quantization using coherent statesFAPESP/CNRS program 2009/54771-2 (2010-2012)

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• 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; semiclassicalmethods 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 interactiontaking into account the interaction with external fields exactly (analogous to the Furry picture inQED 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 scenarioof particle creation in electric field [1, 39, 40], calculations in complicated configurations of externalfields (combinations of electric, magnetic, and plane wave fields) [2, 41, 42, 43], calculations of theparticle 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 externalfields and it was discovered for the first time a close relationship between the creation of particlesin 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 reactionin 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].

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References

[1] V.G. Bagrov, D.M. Gitman and Sh.M. Shvartsman, Concerning the production of electron-positronpairs 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 fromVacuum, 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 andcoherent 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 creatingpairs. 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 externalfield, Journ. Phys. A 10 (1977) 2007-2020.

[8] D.M. Gitman, Processes of arbitrary order in quantum electrodynamics with pair-creating externalfield, 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, PreprintMIT (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 fieldand 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 andits 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, PreprintPhIAN (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 creatingpairs, 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. VUZovFizika (Sov. Phys. Journ.) 7 (1979) 16-21.

[17] S.P. Gavrilov, D.M. Gitman and Sh.M. Shvartsman, The unitarity relation in quantum electrody-namics 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 intensivepair-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-creatingexternal field, Fortschr. Phys. 29 (1981) 381-411.

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

[21] D.M. Gitman and V.A. Kuchin, Generating functional of mean field in quantum electrodynamicswith 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 inpresence of an external field, violating the vacuum stability, Sov. Journal Nucl. Phys. (YadernajaFizika), 46 (1987) 172-180.

[23] E.S. Fradkin, D.M. Gitman and Sh.M. Shvartsman, Optical theorem in quantum electrodynamicswith 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 exter-nal 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 reductionformulas 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 inpresence of an external field, violating the vacuum Stability, Trudy PhIAN (Proceedings of LebedevInstitute, Moscow), 193 (1989) 208-221.

[28] V.P. Barashev, E.S. Fradkin, D.M. Gitman and Sh.M. Shvartsman, The problems of QED withunstable 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 gaugefield, 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 inexternal 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 inexternal 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 onan example of the standard SU(2)×U(1) model, Preprint MIT, CTP # 1995 (1991) 1-53; Problemsof an External Field in Non-Abelian Gauge Theory, Proceedings of the First International SakharovConference on Physics, “Sakharov Memorial Lectures in Physics” Vol.2 (1991) 187-194, Edited byL.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, actingfor 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 electro-magnetic 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 andplane-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 constantuniform 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 principleand 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 inexternal 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 uniforminvariants: 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 ofNeutral 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, ElectromagneticWave Scattering at a Neutral Fermion Possessing Magnetic and Electric Moments, Izw. VUZovFizika 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 thefinite 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 onthe 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 inexternal 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. VUZovFiz. 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 homogeneousfield , 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 inexternal 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 witha 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 electromag-netic field, Proceedings 8th Lomonosov Conference on Elementary Particle Physics (25-30 August1997, 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 withConstant 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 amagnetic 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 inMagnetic Field, in Proceedings of 1996 International Workshop PERSPECTIVES OF STRONGCOUPLING GAUGE THEORIES, (Nagoya, 13-16 November 1996, Japan), Editors J. Nishimuraand 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 FRWUniverse with Electromagnetic Background and some Applications of Them, Proceedings of ForthAlexander 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 back-ground, Phys. Rev. D78, 045017(35) (2008)

[64] S.P. Gavrilov and D.M. Gitman, Energy-momentum tensor in thermal strong-field QED with un-stable 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 inQuantum 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 Presenceof 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 aperfect 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 describedin the Lagrangian and Hamiltonian formulations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. For example, itwas proved for the first time that the number of gauge transformations of the action is equal tothe number of primary first class constraints in the Hamiltonian formulation and that the numberof non-physical variables is equal to the number of first class constraints.

• The Hamiltonization and quantization of singular systems with higher derivatives was formulatedfor 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 zeroHamiltonian systems [14, 15, 16, 17, 18, 19].

• All the symmetries of a general gauge theory were described and, in particular, the form ofgauge transformation was related to the constraint structure of the same theory in Hamiltonianformulation. In this way, it was obtained a rigorous proof of the equivalence of the two definitionsof physical quantities in gauge theories: one of them states that the physical functions are gaugeinvariants in the extremals and the other states that the physical functions are those whichcommute 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 extensionof 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. VUZovFizika (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 Hamiltonianformalisms, In Quantum field theory and quantum statistics v. I, pp. 143—164, Ed. by Batalin, Ishamand Vilkovisky (Adam Hilger, Bristol, 1987).

[7] D.M. Gitman and I.V. Tyutin, Canonical quantization of singular theories, In Group theoreticalmethods 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, June24-29, 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 higherderivatives, 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 withhigher 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 HamiltonianPhenomenon, Proceedings 8th Lomonosov Conference on Elementary Particle Physics (25-30 August1997, 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 HamiltonianPhenomenon, 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,MichaelMarinov Memorial Volume: Multiple Facets of Quantization and Supersymmetry, ed. M. Olshanetskyand 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 gaugesymmetries, J. Phys. A36 (2003) 6587-6609

[23] B. Geyer, D.M. Gitman, and I.V. Tyutin, Reduction of Euler-Lagrange equations in general gaugetheories 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 OFEXTERNAL 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, andphysical 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, U2116 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 ofPhysics, 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,September21-25, 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 ofW3-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 GeneralCoordinates, 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 coor-dinates, 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 anN=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 ap-plication 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 ofMotion, 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-adjointextensions

• It was obtained and systematically investigated for the first time a large number of new classesof 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 differentmethods, for example through the set of exact solutions, Schwinger’s proper time method andpath 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 ofRelativistic Wave Equations [21, 27].

• New exact solutions for the relativistic wave equations in 3+1 and 2+1 dimensions in combinationwith solenoidal Aharonov-Bohm field and some additional electric and magnetic fields were studiedin 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 inphysical 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 ofTwo Classical Plane Waves Propagating in Slightly Different Directions, Soviet Physics Journal 18No. 1 (1975) 34-36)

[2] V.G. Bagrov, D.M. Gitman, P.M. Lavrov, V.N. Zadorozhny and V.N. Shapovalov, New exact solu-tions 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 solu-tions 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 thenull-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 electromag-netic 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 Newexact 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 theDirac 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 exactsolutions 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 quantizedplane-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 electromagneticfields, J. Math. Phys. 23 (1982) 2558-2561.

[18] S.P. Gavrilov, D.M. Gitman, and J.L. Tomazelli, Comments on spin operators and spin-polarizationstates 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 Physik14, 8 (2005) pp. 467-478

19

[20] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, Charged particles in crossed and longitudinal electro-magnetic 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. VUZovRadiofizika (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 PlaneQuantized Monochromatic Electromagnetic Wave, Izw. VUZov Fizika 16 No. 8 (1973) 55-58; (SovietPhysics Journal 16 No. 8 (1973) 1082-1085)

[26] V.G. Bagrov, D.M. Gitman, and V.A. Kuchin, Interaction with an external field in quantumelectrodynamics, 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 andin a Classical Field of Redmond’s Configuration, Izw. VUZov Fizika 17 No.6 (1974) 47-51; (SovietPhysics Journal 17 No. 6 (1974) 787-790)

[28] V.G. Bagrov, D.M. Gitman and P.M. Lavrov, Electron in Constant Crossed Electromagnetic Fieldsand 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 ofQuantized 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 QuantizedPlane Wave traveling in the Same Direction, Izw. VUZov Fizika 17 No. 7 (1974) 60-64; (SovietPhysics 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 electronsinteracting 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-Fieldand 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 inquantized 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 andits 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 andits 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 exter-nal 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 electromagneticfield, 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 constantuniform 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, No5 (1993) 448-452.

[43] D.M. Gitman, Sh.M.Shvartsman and W.da Cruz, Path Integral over Velocities for RelativisticParticle 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 Propagatorin 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 inexternal field, Phys. Rev. D55 (1997) 7701-7714

[46] S.P. Gavrilov and D.M. Gitman, Proper time and path integral representations for the commutationfunction, 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 equationsfor 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 waveequations 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 withmagnetic-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 inmagnetic-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 SciencePublishers, 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 cyclotronand 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 onParticle 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, Angularbehavior 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 inclassical 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 ofpolarization components of synchrotron radiation under changes of particle energy, ”Recent Problemsin 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 ofpolarization components for synchrotron radiation under changes of particle energy, Proceedings ofthe Eleventh Lomonosov Conference on Elementary Particle Physics Particle Physics in Laboratory,Space and Universe, 21-27 August 2003, Moscow, Russia (World Scientific, New Jersey, Singapore2005)

[63] V.G. Bagrov, D.M. Gitman, V.B. Tlyachev, A.T. Jarovoi, New theoretical Results in SynchrotronRadiation, 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-adjointextensions 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-AdjointExtensions of Symmetric Operators II. Differential Operators, Russian Physics Journ. 50No. 9 (2007)853-884

[66] B.L. Voronov, D.M. Gitman, and I.V. Tyutin, Constructing quantum observables and self-adjointextensions 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 Coulombfield, 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 Diracoperators 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 inCalogero problem, J. Phys. A43 (2010) 145205 (34pp)

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

10.4 Path integrals; group theory in relativistic quantum mechanics and field the-ory; semiclassical methods and coherent states

• The path integral method was used in QED with unstable vacuum to calculate the density matrixof particles created by external fields [1, 2]. For the first time, it was constructed a path integralpresenting the generating functional with two sources, which always appears in the quantumtheory 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 tothe case of constrained theories [6, 7].

22

• Different kinds of propagator representations for relativistic particles were constructed throughpath integrals [25, 26, 30, 31, 2, 30, 14, 34, 13, 36]. Two super-generalizations of the Schwingerproper time method were introduced for the first time. For the first time, it was shown that allGrassmann integrations can be performed in the path integral representations for spin particlepropagators, as well as were obtained expressions for the so called spinor factor in arbitrary exter-nal fields [27, 30, 45]. Using the obtained path integral representations, the particle propagatorswere 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 thework [35].

• In the articles [14, 15, 16, 17, 18, 19, 20], coherent states of lie groups and its applications werestudied. 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 inexternal 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 withunstable 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 withunstable vacuum, Preprint PhIAN (Lebedev Institute), 138 (1981).

[4] D.M. Gitman and V.A. Kuchin, Generating functional of mean field in quantum electrodynamicswith 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 inpresence of an external field, violating the vacuum stability, Sov. Journal Nucl. Phys. (YadernajaFizika), 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. VUZovFizika (Sov. Phys. Journ.) 8 (1975) 134-135.

[9] V.G. Bagrov, D.M. Gitman and V.A. Kuchin, Eigenfunctions of linear combinations of creation andannihilation 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 andcoherent 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 anexternal 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 inquantized 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 relativisticparticles in external fields, In Group Theoretical Methods in Physics Proceedings of First ZvenigorodSeminar, (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 quanti-zation 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 andStieltjes 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 rel-ativistic electron in homogeneous magnetic field and in thin solenoid field, Preprint PhIAN (LebedevInstitute), 101 (1986) 1-18.

[24] V.G. Bagrov, D.M. Gitman and V.D. Skarginski, Aharonov-Bohm effect for stationary and coherentstates 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 propagatorsand 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 RelativisticParticle 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 Propagatorin 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 Memorialvol. 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 inexternal field, Phys. Rev. D55 (1997) 7701-7714

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

24

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

[34] D.M. Gitman, Path integrals and pseudoclassical description for spinning particles in arbitrarydimensions, 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 spinlessparticle in large magnetic-solenoid field, Problems of Modern Theoretical Physics (A volumein honour of Professor I.L. Buchbinder in the occasion of his 60th birthday) (Tomsk State UniversityPress, 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-relativisticelectron 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 relativisticelectron in magnetic—solenoid field, to be published

10.5 Classical and pseudoclassical relativistic particle models and their quantiza-tion

• It was presented a generalization for the pseudoclassical action of a spin particle in the presenceof an anomalous magnetic moment [1, 2].

• New pseudoclassical models for relativistic particles were proposed: for a Weyl particle [3]; forspin 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 higherspins (integer and half-integer) in 2+1 dimensions [10, 11] (both supersymmetric models); forDirac Massive particles (spin 1/2) in arbitrary odd dimensions [12, 13].

• It was considered a consistent canonical quantization procedure for pseudoclassical models ofrelativistic spin 1 particles. The constructed quantum mechanics for the massive case proves tobe 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 waspresented by means of path integral [14].

• It was constructed a path integral representation for the commutation function of the quantizedspinorial 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 consistentrelativistic 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.

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[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 spin1 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 DiracParticle, 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 andTheoretical Physics, F.A. Berezin Memorial vol. American Mathematical Society Translation, Series2, 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, inProceedings of SECOND INTERNATIONAL SAKHAROVCONFERENCE 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 inodd 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 arbitrarydimensions, 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 pseudoclassicalmodel 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 ofclassical 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 rela-tivistic 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 RelativisticQuantum Mechanics, Int. J. Mod. Phys. A15 (2000) 4499-4538

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[22] S.P. Gavrilov, D.M. Gitman, Quantization of the Relativistic Particle, Class.Quant.Grav. 17 issue19 (2000) L133-L139

[23] S.P. Gavrilov, D.M. Gitman, Quantization of the Relativistic Particle and Consistent RelativisticQuantum Mechanics, Proceedings of the International Conference ”Quantization, gauge theories, andstrings, 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 particlesand 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-polarizationstates 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 back-ground 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 me-chanics 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 Poincarégroup is considered as a generating function for conventional multicomponent fields. This ap-proach allows an unified consideration to the problem of construction of different kinds of rela-tivistic 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 generalizesWigner’s ideas in relation to the treatment of non-relativistic orientable objects (in particular, anon-relativistic rotator) using two reference frames (one fixed in space and the other fixed in thebody) [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 ofVII 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, inProceedings of SECOND INTERNATIONAL SAKHAROVCONFERENCE 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 pathintegrals, 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

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[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 quan-tized AB—superparticle, Proceedings of Second International Conference ”Quantum Field theory andGravity” (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-LikeDynamical Systems, Mod. Phys. Lett. A13 (1998) 2559-2570

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

[10] D.M. Gitman, and A. Shelepin, Fields on the Poincaré Group: Arbitrary spin description andrelativistic 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 massivespin 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 ExternalGravity,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.Pogosyanand 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 OrientableObjects, 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 com-putation

• It was presented a detailed study of the spin equation , that is, the two levels system describedby two time dependent coupled differential equations. 26 new classes of exact solutions for thosesystems 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 methodfor the differential equation that describes a two levels system. The existence of transformationsunder which the form of the equations of the two levels systems is invariant was demonstrated. Inparticular, a Darboux operator that transforms a problem given by real fields int a new problemalso 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 problemstarting 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 theimportant case of external fields whose effective influence is restrict to a finite time interval [8].

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• Using the exact two and four levels systems, it was described the theoretical implementation of aset of quantum universal logical gates, studying specially the characteristics of the external fieldsand 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 Systemsunder 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 transformationfor 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 underlyinga 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 un-derlying 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 magneticfields restricted in time, Publicação IFUSP ; e-Print: arXiv:0803.0299, submitted to Annalen derPhysik

[9] M.C. Baldiotti, V.G. Bagrov, and D.M. Gitman, Two Interacting Spins in External Fields andApplication 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 θµν is constructed by two bifermionicparameters (Grassmann algebra elements) was introduced. In this approach, the Moyal productcontains a finite number of derivatives, what permits to avoid difficulties in relation to the stan-dard procedure. The renormalizability properties of non-commutative theories of this kind wereanalyzed [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 dy-namics 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 tothe space non-commutativity. The θ-modification of the Pauli equation was constructed. Theθ-modified interaction between non-relativistic spin and the magnetic field was constructed alongwith 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 me-chanics and noncommutative version of Berezin-Marinov action, Europ. Phys. J. C 54 (2008) 325-332

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[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 NoncommutativeParticle 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 inNoncommutative 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 classicaland quantum statistics[1, 3, 5]

• A new form for the Wigner-like distribution functions was proposed to a quantum system inthermal equilibrium. Based in this representation, corrections were calculated to the classicaldistribution.[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 ofthe 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. VUZovFizika (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 functionalon 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 ofa 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 quantummechanics, Izw. VUZov Fizika (Sov. Phys. Journ.) 9 (1971) 98-102.

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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 ofgeneral 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 thetraffic 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, DordrechtBoston London, 2003)

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