Curriculum Vitae Dmitri Maximovitch Guitman (Gitman) · 2 Education and academic titles 1961-1966:...

Curriculum Vitae

Dmitri Maximovitch Guitman (Gitman)

September 6, 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 7

8 Number of publications and international citations 10

9 Coordination of projects and research group 11

10 Summary of the main scienti�c results obtained 11

10.1 Quantum �eld theory with external backgrounds . . . . . . . . . . . . . . . . . . . . . . 1110.2 General theory of constrained systems and their quantization . . . . . . . . . . . . . . . 1610.3 Exact solutions of the relativistic wave equations and theory of self-adjoint extensions . 1810.4 Path integrals; group theory in relativistic quantum mechanics and �eld theory; semi-

classical methods and coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.5 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 �eld 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: [email protected], [email protected]: http://www.dfn.if.usp.br/pesq/tqr/en/gitmanCV Lattes: http://lattes.cnpq.br/7459553192735157Productivity grant PQ-1A of CNPq.

1

2 Education and academic titles

• 1961-1966: Graduation and Master's degree � Department of Physics of Tomsk State University� Russia. Received a medal and the �red diploma�. Master's Thesis: �Integral Equations 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 Quantum

Statistics� � Tomsk State University, Russia. Diploma MFM No. 011435, State Higher Attesta-tion Commission, Moscow, January 16, 1970. Received o�cial 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 Mathemat-

ical Physics. Diploma MDC no. 096071, �Higher Attestation Commission�, Moscow, April 1,1976.

• 1979: Received the title �Doctor in Sciences� , (the highest scienti�c degree in Physics andMathematics given in Russia) with the thesis �Problems of External Fields in Quantum

Electrodynamics� , Institute of Nuclear Physics of Novosibirsk. Diploma FM No. 001066,�Higher Attestation Commission�, Moscow, April 18, 1980. Received o�cial favorable opinionsfrom Professors F.A. Berezin, Moscow State University, B.N. Baier, Institute of Nuclear Physics,Novosibirsk, V.N. Barbashov, International Institute of Nuclear Research, Dubna, and V.I. Manko,Lebedev Physical Institute, Moscow. This title was recognized by University of São Paulo in June5, 1996 as equivalent to Full Professor in accordance with document No. 94.1.1005.43.5

• 1981: Received the academic degree �Cathedratic Professor of the chair of Theoretical

Physics� . Approved by the Russian Ministry of Higher Education. Diploma PR No. 007429,Higher Attestation Commission, Moscow, June 26, 1981.

3 Academic Positions

• 1969-1970: Assistant Professor at the Department of Physics � Tomsk Institute of 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 PedagogicalUniversity (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.

2

4 Didactic Activities

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

In Russia

1. General Physics, 1969-1975, TIASUR

2. Quantum Mechanics, 1970-1975, TIASUR

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

4. Electrodynamics, 1970-1975, TIASUR

5. Mathematical Physics, 1976-1985, TSPU

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

7. Group Theory, 1976-1985, TSPU

8. Quantum Field Theory, 1976-1985, TSU

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

10. General Relativity, 1970-1975, TSU

In Brazil

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

2. Path integrals in quantum mechanics and quantum �eld 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

1. P.V. Bozrikov, Tese de Mestrado: Berson model in QED (Tomsk State University, Tomsk, 1969)

2. P.M. Lavrov. Tese de Mestrado: Exact solvable models in QED (Tomsk State University, Tomsk,1972)

3. V.M. Shachmatov. Tese de Mestrado: Charged particles in strong electromagnetic �elds, (TomskState University, Tomsk, 1974)

1TIASUR-Tomsk Institute of Automation Control Systems and Radio Engineering; TSPU-Tomsk State Pedagogical

Universtity; MIREA-Moscow Institute of Radio Engineering, Electronics and Automation; TSU-Tomsk State University;

USP-University of Sao Paulo

3

4. S.P. Gavrilov. Tese de Mestrado: Particle creation in QED (Tomsk State University, Tomsk,1978)

5. I.M. Lichtzier. Tese de Mestrado: Some quantum processes in external electromagnetic �elds,(Tomsk State University, Tomsk, 1984)

6. V.P. Barashev. Tese de Mestrado: Reduction formulas in QED with unstable vacuum, (TomskState University, Tomsk, 1985)

7. João Luis Meloni Assirati. Dissertação de Mestrado: Generalização covariante do ordenamento

de Weyl e quantização da partícula, (Universidade de São Paulo, São Paulo, Setembro 2001)

8. Mario Cesar Baldiotti, Dissertação de Mestrado: Estados Quânticos de um Elétron em um Campo

Magnético Uniforme, (Universidade de São Paulo, São Paulo, Maio 2002)

9. Rodrigo Fresneda, Dissertação de Mestrado: Quantização da partícula relativística espinorial em

2 + 1 dimensões, (Universidade de São Paulo, São Paulo, Agosto 2003)

10. Tiago Carlos Adorno de Freitas, Dissertação de Mestrado: Particula Espinorial (Pseudo)Classica eQuântica em Espacos Commutativos e Não Comutativos. (Universidade de São Paulo, São Paulo,Agosto 2009)

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 �elds, (TomskState University, Tomsk, 1975);

4. V.M. Shachmatov. Thesis: Quantum processes with a relativistic charged particle, interacting with

strong electromagnetic �eld, (Azerbaijan State University, Baku, 1978);

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

6. I.M. Lichtzier. Thesis: Green's functions and one-loop e�ective action in external gauge and

gravitational �elds, (Tomsk State University, Tomsk, 1988);

7. M.D. Noskov. Thesis: Problems of QED with intensive external �elds, (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 modern

quantum theory, (University of São Paulo, São Paulo, 1998)

13. Jose Nemecio Acosta Jara. Thesis: Quantum theory of particle radiation in a solenoidal magnetic

�eld, (University of São Paulo, São Paulo, December 2002)

4

14. Andrei Smirnov, Thesis: The Dirac equation with a superposition of Aharonov-Bohm and colinear

uniform magnetic �elds, (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 backgrounds

and in non-commutative space-times, (University of São Paulo, São Paulo, October 2008)

17. Vladislav Kupriyanov, Thesis, Quantization of non-Lagrangian systems and non-commutative

quantum mechanics, (University of São Paulo, São Paulo, march 2009)

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

19. Damiao P. Meira Filho, Tese de Doutorado: Movimento quântico e semiclássico em um campo de

um magnético-solenóide, (Universidade de São Paulo, São Paulo, otubro 2010)

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 regular

representation for Poincare group in various dimensions

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

of gauge theories

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

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

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

9. 2010- Nelson Yokomizo, Projeto: Creacao de particulas em campo de Coulomb com Z > 137, emandamento.

6 Books published and in progress

1. Exact Solutions of the Relativistic Wave Equations

with V. G. Bagrov, I. M. Ternov et al.Nauka, Novosibirsk (1982)144 pages (in russian)

5

2. Canonical Quantization of Fields with Constraints

with 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

4. Quantum Electrodynamics with Unstable Vacuum

with E. S. Fradkin e Sh. M. ShvartsmanNauka, Moscow (1991)294 pages (in russian)

5. Quantum Electrodynamics with Unstable Vacuum

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

6. Quantization of Fields with Constraints

with 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.

6

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 the re-spective abstracts:

1. D.M. Gitman and A. Shelepin, Field on Poincaré Group and Quantum Description of Orientable

Objects, Europ. Physical Journal C, 61, Issue1 (2009)111, DOI 10.1140/epjc/s10052-009-0954-x.We propose an approach to the quantum-mechanical description of relativistic orientable objects.It generalizes Wigner's ideas concerning the treatment of nonrelativistic orientable objects (inparticular, a nonrelativistic rotator) with the help of two reference frames (space-�xed and body-�xed). 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 complete setof transformations that test the symmetries of an orientable object and of the embedding spacebelongs 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 �eld f(x, z) is a generatingfunction of usual spin-tensor multicomponent �elds. In the theory under consideration, there arefour di�erent types of spinors, and an orientable object is characterized by ten quantum numbers.We study the corresponding relativistic wave equations and their symmetry properties.

2. S.P. Gavrilov and D.M. Gitman, Consistency Restrictions on Maximal Electric-Field Strength in

Quantum Field Theory, Phys. Rev. Lett. 101, 130403(4) (2008).QFT with an external background can be considered as a consistent model only if backreaction isrelatively small with respect to the background. To �nd the corresponding consistency restrictionson an external electric �eld and its duration in QED and QCD, we analyze the mean energy densityof quantized �elds for an arbitrary constant electric �eld E, acting during a large but �nite 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 �eld. 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 Coulomb

�eld, Theoretical and Mathematical Physics, 150(1) (2007) 34-72 (Translated from Teoretich-eskaya i Matematicheskaya Fizika, 150, No. 1, pp.41-84, 2007).We consider the quantum-mechanical problem of a relativistic Dirac particle moving in the Coulomb�eld of a point charge Ze. In the literature, it is often declared that a quantum-mechanical descrip-tion of such a system does not exist for charge values exceeding the so-called critical charge withZ = α−1 = 137 based on the fact that the standard expression for the lower bound state energyyields complex values at overcritical charges. We show that from the mathematical standpoint,there is no problem in de�ning a self-adjoint Hamiltonian for any value of charge. What is more,the transition through the critical charge does not lead to any qualitative changes in the math-ematical description of the system. A speci�c feature of overcritical charges is a non uniquenessof the self-adjoint Hamiltonian, but this non uniqueness is also characteristic for charge valuesless than the critical one (and larger than the subcritical charge with Z = (

√3/2)α−1 = 118).

We present the spectra and (generalized) eigenfunctions for all self-adjoint Hamiltonians. Themethods used are the methods of the theory of self-adjoint extensions of symmetric operators andthe Krein method of guiding functionals. The relation of the constructed one-particle quantummechanics to the real physics of electrons in superstrong Coulomb �elds where multiparticle e�ectsmay 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).

7

We have obtained non-perturbative one-loop expressions for the mean energy-momentum tensorand current density of Dirac's �eld on a constant electric-like background. One of the goals ofthis calculation is to give a consistent description of back-reaction in such a theory. Two casesof initial states are considered: the vacuum state and the thermal equilibrium state. First, weperform calculations for the vacuum initial state. In the obtained expressions, we separate the con-tributions due to particle creation and vacuum polarization. The latter contributions are relatedto the Heisenberg�Euler Lagrangian. Then, we study the case of the thermal initial state. Here,we separate the contributions due to particle creation, vacuum polarization, and the contributionsdue to the work of the external �eld on the particles at the initial state. All these contributionsare studied in detail, in di�erent regimes of weak and strong �elds and low and high temperatures.The obtained results allow us to establish restrictions on the electric �eld and its duration underwhich QED with a strong constant electric �eld is consistent. Under such restrictions, one canneglect the back-reaction of particles created by the electric �eld. Some of the obtained resultsgeneralize the calculations of Heisenberg�Euler for energy density to the case of arbitrary strongelectric �elds.

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 constraintstructure of a theory in the Hamiltonian formulation. We demonstrate that the symmetry structureof a theory action can be completely revealed by solving the so-called symmetry equation. Wedevelop a corresponding constructive procedure of solving the symmetry equation with the helpof a special orthogonal basis for the constraints. Thus, we succeed in describing all the gaugetransformations of a given action. We �nd 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 �rst-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 ofan arbitrary gauge symmetry, we give a rigorous proof of the equivalence of two de�nitions ofphysicality condition in gauge theories: one of them states that physical functions are gauge-invariant on the extremals, and the other requires that physical functions commute with FCC(the Dirac conjecture).

6. S.P. Gavrilov, D.M. Gitman, and J.L. Tomazelli, Density matrix of a quantum �eld in a particle-

creating background, Nucl. Phys. B 795 [FS] (2008) 645-677.We examine the time evolution of a quantized �eld in external backgrounds that violate thestability of vacuum (particle-creating backgrounds). Our purpose is to study the exact form of the�nal quantum state (the density operator at the �nal 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 �nd a generating functional that allows one to obtain density operatorsfor an arbitrary initial state. Averaging over states of the subsystem of antiparticles (particles), weobtain 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 �nal state of thecomplete system is not in thermal equilibrium. In this respect, we discuss some cases whenpair-creation by an electric-like �eld 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 e�ects of switching the external �eld on ando� are negligible. It is demonstrated that at high temperatures and in slowly varying electric �elds

8

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 de�nition 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 E�ect in cyclotron

and synchrotron radiations, Nucl. Phys. B605 (2001) 425-454.We study the impact of Aharonov-Bohm solenoid on the radiation of a charged particle movingin a constant uniform magnetic �eld. With this aim in view, exact solutions of Klein-Gordonand Dirac equations are found in the magnetic-solenoid �eld. 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 e�ect in the radiation. In particular, itis 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 Relativistic

Quantum Mechanics, Int. J. Mod. Phys. A15 (2000) 4499-4538.We revise the problem of the quantization of relativistic particle models (spinless and spinning),presenting a modi�ed consistent canonical scheme. One of the main point of the modi�cation 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 �eld. In particular, in a physical sector of the Hilbert space acomplete positive spectrum of energies of relativistic particles and antiparticles is reproduced, andall state vectors have only positive norms.

10. D.M. Gitman, Path integrals and pseudoclassical description for spinning particles in arbitrary

dimensions, Nucl. Phys. B 488 (1997) 490-512.The propagator of a spinning particle in external Abelian �eld and in arbitrary dimensions ispresented by means of a path integral. The problem has distinct solutions in even and odd dimen-sions. In even dimensions the representation is just a generalization of one in four dimensions (ithas been known before). In this case a gauge invariant part of the e�ective action in the path inte-gral has a form of the standard (Berezin-Marinov) pseudoclassical action. In odd dimensions thesolution is presented for the �rst time and, in particular, it turns out that the gauge invariant partof the e�ective action di�ers from the standard one. We propose this new action as a candidate todescribe spinning particles in odd dimensions. Studying the hamiltonization of the pseudoclassicaltheory with the new action we show that the operator quantization leads to adequate minimalquantum theory of spinning particles in odd dimensions. Finally the consideration is generalizedfor the case of the particle with anomalous magnetic moment.

11. S.P. Gavrilov and D.M. Gitman, Vacuum instability in external �elds, Phys.Rev.D 53 (1996) 7162-7175.We study particles creation from the vacuum by external electric �elds, in particular, by �elds,which are acting for a �nite time, in the frame of QED in arbitrary space-time dimensions. In allthe cases special sets of exact solutions of Dirac equation (IN- and OUT- solutions) are constructed.Using them, characteristics of the e�ect are calculated. The time and dimensional analysis of thevacuum instability is presented. It is shown that the distributions of particles created by quasi-constant electric �elds can be written in a form which has a thermal character and seams to

9

be universal, i.e. is valid for any theory with quasi-constant external �elds. Its application, forexample, to the particles creation in external constant gravitational �eld reproduces the Hawkingtemperature exactly.

12. D.M. Gitman and S.I. Zlatev, Spin factor in path integral representation for Dirac propagator in

external �eld, Phys. Rev. D55 (1997) 7701-7714.We study the spin factor problem both in 3+1 and 2+1 dimensions which are essentially di�erentfor spin factor construction. Doing all Grassmann integrations in the corresponding path integralrepresentations for Dirac propagator we get representations with spin factor in arbitrary external�eld. Thus, the propagator appears to be presented by means of bosonic path integral only. In 3+1dimensions we present a simple derivation of spin factor avoiding some unnecessary steps in theoriginal brief letter (Gitman, Shvartsman, Phys. Lett. B318 (1993) 122) which themselves needsome additional justi�cation. In this way the meaning of the surprising possibility of completeintegration over Grassmann variables gets clear. In 2+1 dimensions the derivation of the spin factoris completely original. Then we use the representations with spin factor for calculations of thepropagator in some con�gurations of external �elds. Namely, in constant uniform electromagnetic�eld and in its combination with a plane wave �eld.

13. E.S. Fradkin and D.M. Gitman Path integral representation for the relativistic particle propagators

and BFV quantization, Phys. Rev. D 44 (1991) 3230-3236.The path-integral representations for the propagators of scalar and spinor �elds in an externalelectromagnetic �eld 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-creating

external �eld, Fortschr. Phys. 29 (1981) 381-411.In the paper the perturbation theory is constructed for QED, for which the interaction with theexternal pair-creating �eld 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 electromagnetic�eld is built in the case under consideration. E�ective Lagrangian which generates the exactequation for the mean electromagnetic �eld 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 exter-

nal �eld, Journ. Phys. A 10 (1977) 2007-2020.Dyson's perturbation theory analogue for quantum electrodynamic processes with arbitrary initialand �nal states in an external �eld creating pairs has been discussed. The interaction with the�eld is taken into account exactly. The possibility of using Feynman diagrams, together withmodi�ed correspondence rules, for the representation o the above mentioned processes has beendemonstrated.

8 Number of publications and international citations

• 208 complete articles published in journals.

• 26 complete works published in congress annals.

• 17 chapters in published books.

• 6 published books.

10

• Total of 257 publications.

• Comlete "citation index" (according to Google Scholar): 2281.

9 Coordination of projects and research group

• Quantization problems and QED in strong �eldsFAPESP thematic project 1996/07134-8 (1996-2002)

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

• Quantization and problems in quantum �eld 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)

• 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 scienti�c results obtained

Obtained results in the following investigation areas:

• Quantum �eld 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 �eld 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 �eld 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 �eld theory with external backgrounds

• It was elaborated a general formulation of QED with external �elds which violate vacuum stability.In particular, it was constructed a perturbation theory in relation to the radioactive interactiontaking into account the interaction with external �elds 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].

11

• The results obtained for external electromagnetic �elds 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 e�ects were presented. For example, the temporal scenarioof particle creation in electric �eld [1, 39, 40], calculations in complicated con�gurations of external�elds (combinations of electric, magnetic, and plane wave �elds) [2, 41, 42, 43], calculations of theparticle creation e�ects in theories in higher dimensions [40, 47]. It was presented in the articles[44, 45, 46, 28, 62] a general construction of the density matrix of particles created by external�elds and it was discovered for the �rst time a close relationship between the creation of particlesin external electromagnetic �elds and in gravitational �elds [44, 28].

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

• Radioactive processes were calculated in various electromagnetic �elds [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].

References

[1] V.G. Bagrov, D.M. Gitman and Sh.M. Shvartsman, Concerning the production of electron-positron

pairs from vacuum, Zh. Eksp. Teor. Fiz. 68 (1975) 392-399; Sov. Phys.-JETP, Vol. 41, No. 2 (1975)191-194.

[2] V.G. Bagrov, S.P. Gavrilov, D.M. Gitman and Sh.M. Shvartsman, Creation of Boson Pairs from

Vacuum, Izw. VUZov Fizika 18 No. 3 (1975) 71-74; (Soviet Physics Journal 18 NO.3 (1975) 351-354)

[3] V.G. Bagrov, D.M. Gitman and V.A. Kuchin, External �eld in quantum electrodynamics and co-

herent 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 �eld. I. Izw. VUZov Fizika (Sov.Phys. Journ.) 10 (1976) 81-86.

[5] D.M. Gitman, Quantum processes in an intensive electromagnetic �eld. 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 �eld creating

pairs. III. Izw. VUZov Fizika (Sov. Phys. Journ.) I (1977) 94-99.

[7] D.M. Gitman, Processes of arbitrary order in quantum electrodynamics with a pair-creating external

�eld, Journ. Phys. A 10 (1977) 2007-2020.

[8] D.M. Gitman, Processes of arbitrary order in quantum electrodynamics with pair-creating external

�eld, In Quantum electrodynamics with external �eld (Tomsk State University, Tomsk, 1977) pp.132-149.

[9] E.S. Fradkin and D. M. Gitman, Quantum electrodynamics with intense external �eld, PreprintMIT (1978) 1-58.

[10] S.P. Gavrilov, D.M. Gitman and Sh. M. Shvartsman, Green's functions in external electric �eld,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 �eld and

its combination with magnetic �eld and plane-wave �eld, Sov. Journ. Nucl. Phys. (Yadern. Fizika),29 (1979) 1392-1405.

12

[12] S.P. Gavrilov, D.M. Gitman and Sh.M. Shvartsman, Green's functions in external electric �eld and

its combination with magnetic �eld and plane-wave �eld, Kratk. Soob. Fiz. (Lebedev Inst.), No. 2(1979) 22-26.

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

[14] E.S. Fradkin and D.M. Gitman, Problems of quantum electrodynamics with intensive �eld. (Ap-

pendix), Preprint PhIAN (Lebedev Institute), 107 (1979) 1-40.

[15] E.S. Fradkin and D.M. Gitman, Problems of quantum electrodynamics with intensive �eld creating

pairs, Central research inst. for Phys., Budapest, REKI�1979�83, pp. 1-105.

[16] I.L. Buchbinder and D.M. Gitman, A de�nition 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 �eld, Izw. VUZov Fizika (Sov. Phys. Journ. 257-260) 3 (1980)93-96 .

[18] S.P. Gavrilov and D.M. Gitman, Furry picture for scalar quantum electrodynamics with intensive

pair-creating �eld, Izw. VUZov Fizika (Sov. Phys. Journ. 491-496) 6 (1980) 37-42 .

[19] E.S. Fradkin and D.M. Gitman, Furry picture for quantum electrodynamics with pair-creating ex-

ternal �eld, Fortschr. Phys. 29 (1981) 381-411.

[20] I.L. Buchbinder, E.S. Fradkin and D.M. Gitman, Generating functional in quantum �eld theory

with unstable vacuum, Preprint PhIAN (Lebedev Institute), 138 (1981).

[21] D.M. Gitman and V.A. Kuchin, Generating functional of mean �eld in quantum electrodynamics

with unstable vacuum, Izw. VUZov Fizika (Sov. Phys. Journ.) 10 (1981) 80-84.

[22] S.P. Gavrilov, D.M. Gitman and E.S. Fradkin, Quantum electrodynamics at �nite temperature in

presence of an external �eld, 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 electrodynamics

with unstable vacuum, Fortschr. Phys. 36 (1988) 643-669.

[24] V.G. Bagrov, V.P. Barashev, D.M. Gitman, and Sh.M. Shvartsman, Green functions in exter-

nal electromagnetic �eld, in Collection Quantum processes in intense external �elds, pp. 101-111,(Shteentza, Kishenev, 1987) (3B348b88)

[25] V.P. Barashev, D.M. Gitman, E.S. Fradkin and Sh.M. Shvartsman, Peculiarities of reduction for-

mulas 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 �eld,

violating the vacuum stability, Trudy PhIAN (Proceedings of Lebedev Institute, Moscow), 193(1989) 3-207.


presence of an external �eld, 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 with

unstable vacuum. Reduction formulas. The density matrix of particles creating in an external �eld,Trudu PhIAN (Proceedings of Lebedev Institute, Moscow) 201 (1990) 74-94.

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

13

[30] E.S. Fradkin, D.M. Gitman and Sh.M. Shvartsman, Quantum Electrodynamics with Unstable

Vacuum (Springer-Verlag, Berlin Heidelberg New-York London Paris Hong-Kong Barcelona, 1991)pp. 1-300.

[31] D.M. Gitman, E.S. Fradkin and Sh.M. Shvartsman, Quantum Electrodynamics with Unstable

Vacuum , (Nauka, Moscow, 1991) pp. 1�294.

[32] S.P.Gavrilov, D.M.Gitman, Furry Representation for Fermions, interacting with an external gauge

�eld, Izw. VUZov Fizika (Russian Phys. Journ.) No 4 (1995) 102-108.

[33] I.L. Buchbinder and D.M. Gitman, A method of calculation of quantum processes probabilities in

external gravitational �elds. I, Izw. VUZov Fizika (Sov. Phys. Journ.) 3 (1979) 90-95.

[34] I.L. Buchbinder and D.M. Gitman, A method of calculation of quantum processes probabilities in

external gravitational �elds. 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.

[37] S.P. Gavrilov and D.M. Gitman, Problems of an External Field in Non-Abelian Gauge Theory on

an example of the standard SU(2)×U(1) model, Preprint MIT, CTP # 1995 (1991) 1-53; 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, No5 (1993) 448-452.

[39] D.M. Gitman, V.M. Shachmatov and Sh.M. Shvartsman, Pair creation in the electric �eld, acting

for a �nite time, Izw. VUZov Fizika (Sov. Phys. Journ.) 4 (1975) 23-29.

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

[41] V.G. Bagrov, D.M. Gitman and Sh.M. Shvartsman, Pair creation from vacuum by an electromag-

netic �eld 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 �eld and

plane-wave �eld, Izw. VUZov Fizika (Sov. Phys. Journ.) 5 (1981) 108-111.

[43] D.M. Gitman, M.D. Noskov and Sh.M. Shvartsman, Quantum e�ects in a combination of a constant

uniform �eld and a plane wave �eld, Intern. Journ. Mod. Phys. A 6 (1991) 4437�4489.

[44] V.P. Frolov and D.M. Gitman, Density matrix in quantum electrodynamics, equivalence principle

and Hawking e�ect, Journ. Phys. A 15 (1978) 1329-1333.

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

[46] I.L. Buchbinder, D.M. Gitman and V.P. Frolov, Density matrix for particle-creation processes in

external �eld, 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 �eld with space-time uniform

invariants: Exact solutions, Journ. Math. Phys. 39 (1998) 3547-3567

[48] V.G. Bagrov, P.V. Bozrikov, D.M. Gitman, Yu.I. Klimenko and A.I. Khudomjasov, Radiation of

Neutral Fermion with Electric and Magnetic Moments in Constant and Uniform External Electro-

magnetic Fields, Izw. VUZov Fizika 17 No. 6 (1974) 150-151; (Soviet Physics Journal 17 No. 6(1974) 890-891)

14

[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 the

�nite length ondulator, Journ. Technic. Fiz. XLV, 9 (1975) 1948-1953.

[51] V.G. Bagrov, D.M. Gitman and V.N. Rodionov et al., E�ect of a strong electromagnetic wave on

the radiation emitted by weakly excited electrons, moving in magnetic �eld, Zh. Eksp. Teor. Fiz. 71(1976) 433-439.

[52] Yu.Yu. Volfengaut, S.P. Gavrilov, D.M. Gitman and Sh.M. Shvartsman, Radiative processes in

external pair-creating electromagnetic �eld, Sov. Journ. Nucl. Phys. (Yadern. Fizika) 33 (1981)743-757.

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

[54] S.P. Gavrilov and D.M. Gitman, Radiative processes with an electron in constant homogeneous �eld

, Izw. VUZov Fiz. 10 (1982) 102-106 (Sov. Phys. Journ. 10 (1982) 968-972.)

[55] D.M. Gitman, S.D. Odintsov and Yu.I. Shil'nov, Chiral symmetry breaking in d = 3 NJL model in

external gravitational and magnetic �elds, Phys. Rev. D 54, No 4 (1996) 2968-2970

[56] S.P. Gavrilov, D.M. Gitman and S.D. Odintsov, Quantum Scalar Fields in the FRW Universe with

a Constant Electromagnetic Background, Int. J. Mod. Phys. A12 (1997) 4837-4867

[57] S.P. Gavrilov and D.M. Gitman, Quantum processes in FRW Universe with external electromagnetic

�eld, Proceedings 8th Lomonosov Conference on Elementary Particle Physics (25-30 August 1997,Moscow, Russia), Ed. A.I. Studenikin, (Int. Centre for Advanced Studies, Moscow 1999) 105-109

[58] S.P. Gavrilov, D.M. Gitman, and A.E. Gonçalves, Quantum Spinor Field in FRW Universe with

Constant Electromagnetic Background, Int.J.Mod.Phys.A16, No.26 (2001) 4235-4259

[59] I. Brevik, D.M. Gitman and S.D. Odintsov, The e�ective potential of gauged NJL model in a

magnetic �eld, Gravitation and Cosmology, 3 (1997) 100-104

[60] I. Brevik, D.M. Gitman, and S.D. Odintsov, The E�ective Potential of Gauged NJL Model in

Magnetic 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 FRW

Universe with Electromagnetic Background and some Applications of Them, Proceedings of ForthAlexander Friedmann International Seminar on Gravitation and Cosmology, St. Petersburg, Russia,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 �eld 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-�eld 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 in

Quantum Field Theory, Phys. Rev. Lett. 101, 130403(4) (2008)

15

[66] S.P. Gavrilov, D.M. Gitman, On Schwinger Mechanism for Gluon Pair Production in the Presence

of Arbitrary Time Dependent Chromo-Electric Field, Europ. Physical Journal C, 64, Issue 1 (2009)81; DOI: 10.1140/epjc/s10052-009-1135-7

[67] M. Bordag, I.V. Fialkovsky, D.M. Gitman, and D.V. Vassilevich, Casimir interaction between a

perfect conductor and graphene, described by the Dirac model, Phys. Rev. B 80, No.24 (2009)

10.2 General theory of constrained systems and their quantization

• The structure of the physical sector of gauge theories, in general form, was exhaustively describedin the Lagrangian and Hamiltonian formulations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. For example, itwas proved for the �rst time that the number of gauge transformations of the action is equal tothe number of primary �rst class constraints in the Hamiltonian formulation and that the numberof non-physical variables is equal to the number of �rst class constraints.

• The Hamiltonization and quantization of singular systems with higher derivatives was formulatedfor the �rst time [12, 13].

• It was formulated also for the �rst 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 of gaugetransformation was related to the constraint structure of the same theory in Hamiltonian formu-lation. In this way, it was obtained a rigorous proof of the equivalence of the two de�nitions ofphysical quantities in gauge theories: one of them states that the physical functions are gauge in-variants in the extremals and the other states that the physical functions are those which commutewith �rst 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 Hamilto-

nian 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 Hamilto-

nian theory of general form with constraints, Izw. VUZov Fizika (Sov. Phys. Journ.) 3 (1986) 243-250.

[6] D.M. Gitman and I.V. Tyutin, The structure of gauge theories in the Lagrangian and Hamiltonian

formalisms, In Quantum �eld theory and quantum statistics v. I, pp. 143�164, Ed. by Batalin, Ishamand Vilkovisky (Adam Hilger, Bristol, 1987).

16

[7] D.M. Gitman and I.V. Tyutin, Canonical quantization of singular theories, In Group theoretical

methods in physics Proceedings of Second Zvenigorod Seminar Group Theoretical Methods Physics,Zvenigorod, USSR, 1988, pp. 207-237.

[8] D.M. Gitman, P.M. Lavrov and I.V. Tyutin, Non-point transformation in constrained theories, Jour-nal 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 Coordinates,Proceedings of 3-rd International Sakharov Conference on Physics, Moscow, Russia, June 24-29, 2002,Vol.II, Editors A. Semikhatov, M. Vasiliev, V. Zaikin (Scienti�c World Publ. 2003) pp. 54-63

[12] D.M. Gitman, S.L. Ljachovich and I.V. Tyutin, Hamiltonian formalism of theories with higher

derivatives, Izw. VUZov Fizika (Sov. Phys. Journ. 730-735) 8 (1983) 61-66.

[13] D.M. Gitman, S.L. Ljachovich and I.V. Tyutin, Canonical quantization of Yang-Mills theory with

higher derivatives, Izw. VUZov Fizika (Sov. Phys. Journ.) 7 (1985) 37-40.

[14] D.M. Gitman and I.V. Tyutin, Canonical quantization of �elds with constraints (Nauka,Moscow, 1986) pp. 1-216.

[15] D.M. Gitman and I.V. Tyutin, Quantization of Fields with Constraints pp. 1�291 (Springer-Verlag, Berlin Heidelberg New-York London Paris Hong-Kong Barcelona, 1990).

[16] S.P. Gavrilov and D.M. Gitman, Quantization of Systems with Time-Dependent Constraints. Ex-

ample of Relativistic Particle in Plane Wave, Class. Quantum Grav. 10 (1993) 57-67.

[17] G. Fulop, D.M. Gitman and I.V. Tyutin, Reparametrization Invariance as Gauge Symmetry, Int.J. Theor. Phys. 38 (1999) 1953-1980

[18] G. Fulop, D.M. Gitman, and I.V. Tyutin, Reparametrization Invariance and Zero Hamiltonian

Phenomenon, Proceedings 8th Lomonosov Conference on Elementary Particle Physics (25-30 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 Hamiltonian

Phenomenon, in Topics in Theoretical Physics II, Festschrift for Abraham H. Zimerman Ed. H.Aratyn, L.Ferreira, J. Gomes, (IFT/UNESP-São Paulo-SP-Brazil-1998) pp. 286-295

[20] D. M. Gitman, and I.V. Tyutin, Constraint reorganization consistent with Dirac procedure, 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 gauge

symmetries, J. Phys. A36 (2003) 6587-6609

[23] B. Geyer, D.M. Gitman, and I.V. Tyutin, Reduction of Euler-Lagrange equations in general gauge

theories with external �elds, 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.

17

[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 modi�ed 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 Symmetry,Proceedings of International Seminar Physical Variables in Gauge Theories, Dubna,September 21-25,Russia, 1999, Ed. by A.Khvedelidze, M.Lavelle, D.McMullan, and V.Pervushin, Dubna, 2000, pp.118-128

[31] B. Geyer, D.M. Gitman, P. Lavrov, P. Moshin, On Problems of the Lagrangian Quantization of

W3-gravity, Int.J.Mod.Phys. A18, No.27 (2003) 5099-5125

[32] B. Geyer, D.M. Gitman, P. Lavrov, P. Moshin, Super�eld Extended BRST Quantization in General

Coordinates, Int.J.Mod.Phys.A19 (2004) pp.737-750

[33] D.M. Gitman, P.Yu. Moshin, J.L. Tomazelli, On super�eld covariant quantization in general coor-

dinates, Eur. Phys. J. C44 (2005) 591-598

[34] D.M. Gitman, P.Yu. Moshin, A.A. Reshetnyak, Local Super�eld Lagrangian BRST Quantization,

J. Math. Phys. 46:072302 (2005)

[35] D.M. Gitman, P.Yu. Moshin, and A.A. Reshetnyak, An embedding of the BV quantization into an

N=1 local super�eld 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 super�eld 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 of

Motion, Journal of Math. Sciences 141 (2007) 1399-1406

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

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

10.3 Exact solutions of the relativistic wave equations and theory of self-adjoint

extensions

• It was obtained and systematically investigated for the �rst time a large number of new classes ofexact solutions for relativistic wave equations in external �elds [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].

18

• Various species of Green functions for relativistic wave equations were calculated with di�erentmethods, 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 of

Relativistic Wave Equations [21, 27].

• New exact solutions for the relativistic wave equations in 3+1 and 2+1 dimensions in combinationwith solenoidal Aharonov-Bohm �eld and some additional electric and magnetic �elds were studiedin detail [48, 49, 50, 51, 52, 53, 54]. The obtained solutions were used in the study of the Aharonov-Bohm e�ect under the correspondent electromagnetic �elds [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].

References

[1] V.G. Bagrov, D.M. Gitman, V.N. Zadorozhny and Sh.M. Shvartsman, An Electron in the Field of

Two Classical Plane Waves Propagating in Slightly Di�erent 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 the

null-plane of one class of solutions of relativistic wave equations, Izw. VUZov Fizika (Sov. Phys.Journ.) 8 (1975) 43-49.

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

[7] V.G. Bagrov, D.M. Gitman and A.V. Jushin, Solutions for the motion of an electron in electromag-

netic �eld, Phys. Rev. D 12 (1975) 3200-3201.

[8] V.G. Bagrov, D.M. Gitman, A.G. Meshkov and V. N. Shapovalov, Supplement to the work New

exact solutions of the Dirac equation. II, III Izw. VUZov Fizika (Sov. Phys. Journ.) 1 (1977) 126-127.

[9] V.G. Bagrov, D.M. Gitman, A.V. Shapovalov and V.N. Shapovalov, New exact solutions of the Dirac

equation. VI Izw. VUZov Fizika (Sov. Phys. Journ.) 6 (1977) 105-114.

[10] V.G. Bagrov, D.M. Gitman, N.B. Suchomlin et al., New exact solutions of the Dirac equation. VII.

Izw. VUZov Fizika (Sov. Phys. Journ.) 7 (1977) 46-51.

[11] V.G. Bagrov, D.M. Gitman, P.M. Lavrov and V.N. Zadorozhni, Characteristic features of exact

solutions of the problem of an electron in quantized �eld of a plane wave, Izw. VUZov Fizika (Sov.Phys. Journ.) 3 (1977) 7-14.

[12] V.G. Bagrov, D.M. Gitman and A. V. Shapovalov, Integrals of motion in the electron in quantized

plane-wave problem, Izw. VUZov Fizika (Sov. Phys. Journ.) 2 (1977) 116-121.

[13] V.G. Bagrov and D.M. Gitman, Exact solutions of the relativistic wave-equations in external �eld,In Quantum electrodynamics with external �elds, (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.

19

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

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

[17] V.G. Bagrov, D.M. Gitman and V.N. Shapovalov, Electron motion in longitudinal electromagnetic

�elds, J. Math. Phys. 23 (1982) 2558-2561.

[18] S.P. Gavrilov, D.M. Gitman, and J.L. Tomazelli, Comments on spin operators and spin-polarization

states of 2 + 1 fermions, Eur. Phys. J. C (2005) DOI: 10.1140/epjc/s2004-02026-9

[19] V.G. Bagrov, D.M. Gitman, Non-Volkov solutions for a charge in a plane wave, Annalen der Physik14, 8 (2005) pp. 467-478

[20] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, Charged particles in crossed and longitudinal electro-

magnetic �elds 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 �eld, Izw. VUZovRadio�zika (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 �eld, Theor. Mat. Fiz., 14 2 (1973) 202-210.

[25] V.G. Bagrov, P.V. Bozrikov, D.M. Gitman and P.M. Lavrov, An Electron in a Field of a Plane

Quantized Monochromatic Electromagnetic Wave, Izw. VUZov Fizika 16 No. 8 (1973) 55-58; (SovietPhysics Journal 16 No. 8 (1973) 1082-1085)

[26] V.G. Bagrov, D.M. Gitman, and V.A. Kuchin, Interaction with an external �eld in quantum elec-

trodynamics, Izw. VUZov Fizika (Sov. Phys. Journ.) 4 (1974) 152-153.

[27] V.G. Bagrov, D.M. Gitman and P.M. Lavrov, Electron in a Quantized Field of a Plane Wave and

in a Classical Field of Redmond's Con�guration, 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 Fields

and Plane-Wave Fields, Izw. VUZov Fizika 17 NO.6 (1974) 68-74; (Soviet Physics Journal 17 No.6(1974) 806-811)

[29] V.G. Bagrov, P.V. Bozrikov and D.M. Gitman, Fermion with Anomalous Moment in a Field of

Quantized Plane Wave, Izw. VUZov Fizika 17 No. 6 (1974) 129-132; (Soviet Physics Journal 17 No.6(1974) 864-866)

[30] V.G. Bagrov, D.M. Gitman and V.A. Kuchin, Electron in the Field of Classical and a Quantized

Plane Wave traveling in the Same Direction, Izw. VUZov Fizika 17 No. 7 (1974) 60-64; (Soviet PhysicsJournal 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 �eld. 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 �eld. II, Izw. VUZov Fizika (Sov. Phys. Journ.) 7 (1975) 11-15.

[33] V.G. Bagrov, D.M. Gitman and Sh.M. Shvartsman, Electron in a Quantized Plane Wave-Field and

the Classical Field of a Longitudinal Electric Wave, Izw. VUZov Fizika 18 No. 3 (1975) 67-71; (SovietPhysics Journal 18 No. 3 (1975) 374-350)

20

[34] V.G. Bagrov, I.L. Buchbinder, D.M. Gitman and P.M. Lavrov, Coherent states of the electron in

quantized electromagnetic wave, Theor. Mat. Fiz. 33 (1977) 419-426.

[35] S.P. Gavrilov, D.M. Gitman and Sh. M. Shvartsman, Green's functions in external electric �eld,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 �eld and

its combination with magnetic �eld and plane-wave �eld, 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 �eld and

its combination with magnetic �eld and plane-wave �eld, Kratk. Soob. Fiz. (Lebedev Inst.), No. 2(1979) 22-26.

[38] E.S. Fradkin and D.M. Gitman, Problems of quantum electrodynamics with intensive �eld. (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 �eld, in Collection Quantum processes in intense external �elds, pp. 101-111,(Shteentza, Kishenev, 1987) (3B348b88)

[40] D.M. Gitman, M.D. Noskov and Sh.M. Shvartsman, Green's functions in external electromagnetic

�eld, Izw. VUZov Fizika (Sov. Phys. Journ.) 5 (1989) 59-64.

[41] D.M. Gitman, M.D. Noskov and Sh.M. Shvartsman, Quantum e�ects in a combination of a constant

uniform �eld and a plane wave �eld, 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 Relativistic Par-

ticle Propagator, Bras. Journ. Phys. 24, No.4 (1994) 844-854.

[44] D.M. Gitman, S.I. Zlatev and W.da Cruz, Spin Factor and Spinor Structure of Dirac Propagator

in Constant Field, Bras. Journ. Phys. 26 (1996) 419-425

[45] D.M. Gitman and S.I. Zlatev, Spin factor in path integral representation for Dirac propagator in

external �eld, 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 �eld in combination with other electromagnetic �elds, 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 �elds, J. Math. Phys. 42, No.5 (2001)

[50] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, and I.V. Shirokov, New solutions of relativistic wave

equations in magnetic �eld and longitudinal �elds, J.Math. Phys. 43 (2002) 2284-2295

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

[52] S.P. Gavrilov, D.M. Gitman, and A.A. Smirnov, Green functions of the Dirac equation with

magnetic-solenoid �eld, J. Math. Phys. 45 (2004) 1873

21

[53] S.P. Gavrilov, D.M. Gitman, and A.A. Smirnov, Self-adjoint extensions of Dirac Hamiltonian in

magnetic-solenoid �eld 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 �eld, �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 E�ect in cyclotron and

synchrotron radiations, Publicação IFUSP 1395/2000; hep-th/0001108; quant-ph/001022, Nucl. Phys.B605 (2001) 425-454

[56] V.G. Bagrov, D.M. Gitman, A. Levin, and V.B. Tlyachev, Impact of Aharonov-Bohm Solenoid on

Particle Radiation in Magnetic Field, Mod.Phys.Lett. A16, No. 18 (2001) 1171-1179

[57] V.G. Bagrov, D.M. Gitman, and V.B. Tlyachev, l-dependence of particle radiation in magnetic-

solenoid �eld and Aharonov-Bohm E�ect, Int. Journ. Mod. Phys. A17 (2002) 1045-1048

[58] V.G. Bagrov, D.M. Gitman, and V.B. Tlyachev, Aharonov-Bohm E�ect 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 in clas-

sical theory of synchrotron radiation, Surface. Roentgen, synchrotron, and neutron studies. No. 11(2003) pp. 59-65.

[61] V.G. Bagrov, D.M. Gitman, V.B. Tlyachev, and A.T. Jarovoi, Evolution of angular distribution of

polarization components of synchrotron radiation under changes of particle energy, �Recent 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 of

polarization 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 Scienti�c, New Jersey, Singapore2005)

[63] V.G. Bagrov, D.M. Gitman, V.B. Tlyachev, A.T. Jarovoi, New theoretical Results in Synchrotron

Radiation, Nuclear Instruments & Methods in Physics Research, B240 (2005) pp 638 - 645

[64] B.L. Voronov, D.M. Gitman, and I.V. Tyutin, Constructing quantum observables and self-adjoint

extensions of symmetric operators I, Russian Phys. Journ. No. 1 (2007) 1-31

[65] B.L. Voronov, D.M. Gitman, and I.V. Tyutin, Constructing Quantum Observables and Self-Adjoint

Extensions of Symmetric Operators II. Di�erential Operators, Russian Physics Journ. 50 No. 9 (2007)853-884

[66] B.L. Voronov, D.M. Gitman, and I.V. Tyutin, Constructing quantum observables and self-adjoint

extensions of symmetric operators III. Self-adjoint boundary conditions, Izv. Vuzov Fizika, 51, No. 2(2008) 3-43 (in Russian); Russian Phys. Journ. 51, No. 2 (2008) 115-157 (English translation)

[67] B.L. Voronov, D.M. Gitman, and I.V. Tyutin, The Dirac Hamiltonian with a superstrong Coulomb

�eld, Theoretical and Mathematical Physics, 150(1) (2007) 34-72

[68] D.M. Gitman, A. Smirnov, I.V. Tyutin, and B.L. Voronov, Self-adjoint Schrödinger and Dirac

operators with Aharonov-Bohm and magnetic-solenoid �elds, arXiv:0911.0946v1 [quant-ph]

[69] D.M. Gitman, I.V. Tyutin, and B.L. Voronov, Self-adjoint extensions and spectral analysis in

Calogero problem, J. Phys. A43 (2010) 145205 (34pp)

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

22

10.4 Path integrals; group theory in relativistic quantum mechanics and �eld 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 �elds [1, 2]. For the �rst time, it was constructed a path integralpresenting the generating functional with two sources, which always appears in the quantum theoryof �elds 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].

• Di�erent 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 �rst time. For the �rst time, it was shown thatall Grassmann 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 external�elds [27, 30, 45]. Using the obtained path integral representations, the particle propagators werecalculated in various con�gurations of external �elds [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 in

external �eld, Izw. VUZov Fizika (Sov. Phys. Journ. 529-533) 6 (1980) 77-81.

[2] V.P. Barashev, E.S. Fradkin, D.M. Gitman and Sh.M. Shvartsman, The problems of QED with

unstable vacuum. Reduction formulas. The density matrix of particles creating in an external �eld,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 �eld theory with

unstable vacuum, Preprint PhIAN (Lebedev Institute), 138 (1981).

[4] D.M. Gitman and V.A. Kuchin, Generating functional of mean �eld in quantum electrodynamics

with unstable vacuum, Izw. VUZov Fizika (Sov. Phys. Journ.) 10 (1981) 80-84.


presence of an external �eld, violating the vacuum stability, Sov. Journal Nucl. Phys. (YadernajaFizika), 46 (1987) 172-180.

[6] D.M. Gitman and I.V. Tyutin, Canonical quantization of �elds 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 and

annihilation operators, Izw. VUZov Fizika (Sov. Phys. Journ.) 9 (1975) 13-19.

[10] V.G. Bagrov, D.M. Gitman and V.A. Kuchin, External �eld in quantum electrodynamics and co-

herent states In Actual problems of theoretical physics (Moscow State University, Moscow, 1976) pp.334-342.

23

[11] V.G. Bagrov, I.L. Buchbinder and D.M. Gitman, Coherent states of a relativistic particle in an

external electromagnetic �eld, Journ. Phys. A 9 (1976) 1955-1965.

[12] V.G. Bagrov, I.L. Buchbinder, D.M. Gitman and P.M. Lavrov, Coherent states of the electron in

quantized electromagnetic wave, Theor. Mat. Fiz. 33 (1977) 419-426.

[13] V.G. Bagrov, I.L. Buchbinder and D.M. Gitman, Construction of coherent states for relativistic

particles in external �elds, 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.

[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 �eld and

Stieltjes moment problem, Physics Letters A 373 (2009) 1916-1920

[22] D.M. Gitman and A.L. Shelepin, Representations of SU(N) groups on the polynomials of the anti-

commuting 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 e�ect for quantum states of

relativistic electron in homogeneous magnetic �eld and in thin solenoid �eld, Preprint PhIAN (LebedevInstitute), 101 (1986) 1-18.

[24] V.G. Bagrov, D.M. Gitman and V.D. Skarginski, Aharonov-Bohm e�ect for stationary and coherent

states of an electron in homogeneous magnetic �eld, Trudy PhIAN (Proceedings of Lebedev Institute,Moscow), 176 (1986) 151-166.

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

[26] E.S. Fradkin and D.M. Gitman Path integral representation for the relativistic particle propagators

and BFV quantization, Phys. Rev. D 44 (1991) 3230-3236.

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



[29] D.M. Gitman, S.I. Zlatev and W.da Cruz, Spin Factor and Spinor Structure of Dirac Propagator

in Constant Field, Bras. Journ. Phys. 26 (1996) 419-425

24

[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 in

external �eld, 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

[33] D.M. Gitman, Path integrals and pseudoclassical description for spinning particles in arbitrary

dimensions, Nucl. Phys. B 488 (1997) 490-512


dimensions, in Functional Integration: Basics and Applications, Ed. C. DeWitt-Morette, P. Cartier,and A. Folacci, NATO ASI Series B: Physics Vol.361, p 418, (Plenum Publishing Corp. 1997)

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

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

[37] V.G. Bagrov, S.P. Gavrilov, D.M. Gitman, and D. P. Meira Filho, Coherent states of spinless

particle in large magnetic-solenoid �eld, 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 �eld, 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 �eld, 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]; forspin 1 massive particles (Chern-Simons particles) [9] and for massive particles with higher spins(integer and half-integer) in 2+1 dimensions [10, 11] (both supersymmetric models); for DiracMassive 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 �elds in arbitrary dimensions waspresented by means of path integral [14].

• It was constructed a path integral representation for the commutation function of the quantizedspinorial �eld interacting with arbitrary external electromagnetic �elds [46].

• For the �rst 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 �rst time a consistentrelativistic quantum mechanics was constructed.

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

25

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.

[3] D.M. Gitman, A.E. Gonçalves and I.V. Tyutin, New pseudoclassical model for Weyl particles, Phys.Rev. D 50 (1994) 5439-5442.



[5] D.M. Gitman, A.E. Gonçalves and I.V. Tyutin, Quantization of a pseudoclassical model of the spin

1 relativistic particle, Int.J.Mod.Phys. A 10 (1995) 701-718.

[6] D.M. Gitman, A.E. Gonçalves and I.V. Tyutin, Pseudoclassical Supergauge Model for 2+1 Dirac

Particle, Physics of Atomic Nuclei, 60 No.4 (1997) 748-752

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

[8] D.M. Gitman, Pseudoclassical Theory of Relativistic Spinning Particle, in Topics in Statistical and

Theoretical Physics, F.A. Berezin Memorial vol. American Mathematical Society Translation, 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 SAKHAROV CONFERENCE ON PHYSICS, Moscow,Russia, 20-24 May 1996, ed. I.M. Dremin, A.M. Semikhatov, (World Sci. Singapore, 1997) 428-434

[12] D.M. Gitman and A.E. Gonçalves, Pseudoclassical description of the massive Dirac particles in odd

dimensions, Int. J. Theor.Phys. 35 (1996) 2427-2438

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


dimensions, Nucl. Phys. B 488 (1997) 490-512

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

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

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

[18] D.M. Gitman, A.E. Gonçalves and I.V. Tyutin, Remark to the Comment on �New pseudoclassical

model for Weyl particles, Preprint FIAN/TD/96-03; hep-th/9602151

[19] D. M. Gitman, and I.V. Tyutin, A pseudo-classical model of a Weyl particles and quantization of

classical constants, Russian Physics Journal 45, No.7 (2002) pp. 690-694

26

[20] A.A. Deriglazov and D.M. Gitman, Classical description of spinning degrees of freedom of relativistic

particles by means of commuting spinors, Publicação IFUSP 1324/98; hep-th/9811229; Mod. Phys.Lett. A14 (1999) pp. 709-720

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

[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 Relativistic

Quantum Mechanics, Proceedings of the International Conference �Quantization, gauge theories, andstrings, Moscow, Russia, June 5-10, 2000� dedicated to the memory of Professor E�m Fradkin, Ed.A. Semikhatov, M. Vasiliev, V. Zaikin, v.II (Scienti�c World, 2001) pp.27-35.

[24] S.P. Gavrilov, D.M. Gitman, Quantization of a spinning particle in an arbitrary background,Class.Quant.Grav. 18 (2001) 2989-2998

[25] R. Fresneda, S. Gavrilov, D. Gitman, and P. Moshin, Quantization of ( 2 + 1)-spinning particles

and bifermionic constraint problem, Class.Quant.Grav.21 (2004) pp.1419-1442

[26] S.P. Gavrilov, D.M. Gitman, and J.L. Tomazelli, Comments on spin operators and spin-polarization

states of 2 + 1 fermions, Eur. Phys. J. C (2005) DOI: 10.1140/epjc/s2004-02026-9

[27] D.M. Gitman, Berezin-Marinov's pseudoclassical action, �Reminiscences about Felix Berezin-

founder of supermathematics�, ed. by E. Karpel, P.A. Minlos, I.V. Tyutin, and D.A. Leites, and(M(TZ)NMO, Moscow 2009) pp. 139-148; ISBN 978-5-94057-458-3

[28] R. Fresneda and D. Gitman, Pseudoclassical description of scalar particle in non-Abelian background

and path-integral representations, Intern. Journ. Mod. Phys. A 23 (6) (2008) 835-853.

[29] D. Gitman and V.G. Kupriyanov, Path integral representations in noncommutative quantum 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 �eld under the Poincarégroup is considered as a generating function for conventional multicomponent �elds. This approachallows an uni�ed consideration to the problem of construction of di�erent kinds of relativistic waveequations and justi�es 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 �xed in space and the other �xed 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 of VIIInternational 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 SAKHAROV CONFERENCE ON PHYSICS, Moscow,Russia, 20-24 May 1996, ed. I.M. Dremin, A.M. Semikhatov, (World Sci. Singapore, 1997) 428-434

27

[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 super-

string 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

[6] A.A. Deriglazov, Galajinsky, and D.M. Gitman, On zero modes of the eleven dimensional superstring,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 �rst 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-Like

Dynamical Systems, Mod. Phys. Lett. A13 (1998) 2559-2570

[9] A.V. Galajinsky and D.M. Gitman, On minimal coupling of the ABC-superparticle to supergravity

background, Phys. Rev. D59 (1999) 047504

[10] D.M. Gitman, and A. Shelepin, Fields on the Poincaré Group: Arbitrary spin description and

relativistic wave equations, Int.J.Theor.Phys. 40 (2001) 603-684

[11] I.L. Buchbinder, D.M. Gitman, V.A. Krykhtin, and V.D. Pershin, Equations of motion for massive

spin 2 �eld coupled to gravity, Nucl.Phys. B584 No.1-2 (2000) 615-640

[12] I.L. Buchbinder, D.M. Gitman, and V.D. Pershin, Causality of Massive Spin 2 Field in External

Gravity,Phys.Lett.B492 (2000) 161-170

[13] D.M. Gitman, and A. Shelepin, Z-description of the relativistic spin, Proceedings of XXIII Interna-tional Colloquium on Group Theoretical Methods in Physics, Edited by A.N.Sissakian, G.S.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 automor-phisms of the proper Poincare group, Int. J. Theor. Phys. 41, No. 4 (2002) 753-790

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

[16] D.M. Gitman and A. Shelepin, Field on Poincaré Group and Quantum Description of Orientable

Objects, Europ. Physical Journal C, 61, Issue1 (2009)111

[17] D.M. Gitman and A.L. Shelepin, Classi�cation 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 di�erential 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 di�erential 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 �elds int a new problemalso given by real �elds was constructed. [2, 3].

28

• 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 �elds whose e�ective in�uence is restrict to a �nite time interval [8].

• 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 �eldsand the possible practical scenarios for the implementations of such devices [7, 9].

References

[1] V.G. Bagrov, J.C.A. Barata, D.M. Gitman, and W.F. Wreszinski, Aspects of Two-Level Systems

under External Time Dependent Fields , J. Phys. A34 (2001) 10869-10879

[2] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, and V.V. Shamshutdinova, Darboux transformation for

two-level system, Annalen der Physik 14 (2005) 390-397

[3] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, and A.D. Levin, Spin equation and its solutions, Annalender 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 �eld, 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 �eld.

Four-level systems, Annalen der Physik, 16, 8 (2007) 274-285

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

[8] V.G. Bagrov, M.C. Baldiotti, D.M. Gitman, and A.D. Levin, Two and four-level systems in magnetic

�elds 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 and

Application to Quantum Computation, Physics of Particles and Nuclei Letters, 6, No. 7 (2009) pp.559-562

10.8 Quantum mechanics and �eld 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 �nite number of derivatives, what permits to avoid di�culties 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-modi�ed actions (modi�ed by the coordinate non-commutativity)for the relativistic scalar and spinorial particles was discussed. The classical and quantum dynam-ics of a charged particle in non-commutative spaces was considered [2, 4, 5].

• It was analyzed the modi�cation of the energy levels of the relativistic hydrogen atom due tothe space non-commutativity. The θ-modi�cation of the Pauli equation was constructed. Theθ-modi�ed interaction between non-relativistic spin and the magnetic �eld was constructed alongwith a θ-modi�cation of the Heisenberg model for coupled spins [6, 7].

29

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 mechan-

ics and noncommutative version of Berezin-Marinov action, Europ. Phys. J. C 54 (2008) 325-332

[3] R. Fresneda, D.M. Gitman, and D.V. Vassilevich, Nilpotent noncommutativity and renormalization,Phys. Rev. D78: 025004 (2008)

[4] D.M. Gitman and V.G. Kupriyanov, Gauge Invariance and Classical Dynamics of Noncommutative

Particle Theory, Journal of Mathematical Physics, 51, 022905 (2010) 022905 (1-8)

[5] J.P. Gazeau, M.C. Baldiotti, and D.M. Gitman, Semiclassical and quantum motion on non-

commutative plane, Physics Letters A, DOI information: 10.1016/j.physleta.2009.08.059

[6] T.C. Adorno, M.C. Baldiotti, M. Chaichian, D.M. Gitman, and A. Tureanu, Dirac Equation in

Noncommutative Space for Hydrogen Atom, Phys. Letters B 682, Issue 2, (2009) 235-239

[7] T.C. Adorno, M.C. Baldiotti, and D.M. Gitman, Quantum and pseudoclassical description of non-

relativistic spinning particles in noncommutative space, submitted to Physics Lett. B

10.9 Quantum Statistics

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

• New kinds of integral-di�erential 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 �rst works in which the idea of e�ective action was introduced and in which, in particular,was demonstrated in detail the case of the quantum statistics.

References

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

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

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

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

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

[6] D.M. Gitman, An expression for the thermodynamical potential in the form of a stationary functional

on mean occupation numbers, Izw. VUZov Fizika (Sov. Phys. Journ.) 4 (1970) 96-102.

[7] D.M. Gitman and A.G. Tchernishov, A variational principle for the thermodynamical potential of a

two-component system Izw. VUZov Fizika (Sov. Phys. Journ.) 3 (1971) 30-35

30

[8] E.A. Arinshtein and D.M. Gitman, Equations for generating functional in classical and quantum

mechanics, Izw. VUZov Fizika (Sov. Phys. Journ.) 9 (1971) 98-102.

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 Scienti�c, Singapore 2007) pp 181-205; russian translation in�Reminiscences about Felix Berezin-founder of supermathematics�, ed. by E. Karpel, P.A.Minlos, I.V. Tyutin, and D.A. Leites, and (M(TZ)NMO, Moscow 2009) pp. 282-301

[2] D.M. Gitman, I.V. Tyutin,J.L. Assirati, and M.G. da Costa, Structure of Lorentz transformation of

general form, Gravitation and Cosmology 4 No.2(14) (1998) 163-166

[3] E.R. Berdichevkaja, S.P. Gavrilov and D.M. Gitman, A mathematical model for calculation of the

tra�c 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 SUPER-

SYMMETRY� , Eds. S. Duplij, W.Siegel, J.Bagger, (Kluwer Acad. Publisher, Dordrecht BostonLondon, 2003)

31

Curriculum Vitae Dmitri Maximovitch Guitman (Gitman) · 2 Education and academic titles 1961-1966:...

Documents

Transcript of Curriculum Vitae Dmitri Maximovitch Guitman (Gitman) · 2 Education and academic titles 1961-1966:...