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At May 21, 1996 Jerzy Lukierski, ordinary professor at the Institute of Theoretical Physics of the University of Wroclaw, will be sixty years old. His friends and colleagues dedicate this volume as their homage to him.
As far as I am concerned I know Jerzy Lukierski since approximately 40 years; the first time I met him when he was still a student of the University. He impressed me in the course of years when our friendship grew by his nonstandard and straightforward behaviour and warm personality.
Jerzy Lukierski got his Ph.D. at our University in May 1961 for his dissertation
Professor Lukierski is an outstanding scientist, well known specialist in the theory of fields as well as the group-theoretical and geometrical foundations of the theory of fundamental interactions. His work is characterized by original, brilliant ideas, close links with the up-to-date scientific developments and the importance of considered problems. The most significant achievement in his scientific career is in my opinion - the first in scientific literature model of quantum deformations of the Poincaré algebra, invented by him and his collaborators in 1991-92. For this achievement he got the award of the Polish Minister of National Education in 1993 and the Maria Sklodowska-Curie award in 1995. At present Lukierski is in the first line of scientists working in these topics and has been leading a large international group of researchers. This group has been exploring the consequences of the theory of quantum groups in describing the four-dimensional symmetries of space-time as well as in dynamical physical theories, i.e. classical mechanics, quantum mechanics and field theory.
But also his earlier results carry a considerable scientific weight. Let me mention some of them, the most important ones.
1. In his work in 1967-77 Lukierski investigated the description of the four-dimensional Green functions in the quantum field theory [50-54, 56] and the formulation of the renormalization group in terms of quantized field operators [76, 81-83]. He considered the distributional structure of Green functions in the equal time limit and explored the behaviour of the commutator of the renormalized fields for small intervals of time. These results were exploited in the definition of the renormalization constants as singularities of the Green functions. Also the transformations of renormalization group described by Callan-Symanzik equation were given by him in terms of the renormalized field operators.
2. In the same period of time, 1967-78, Lukierski proposed the general theory [48, 49, 60, 79] of non-stable particles and resonances in quantum field theory and explored their models [62, 63, 70, 79, 90]. He gave also the generalized formalism of the scattering matrix for resonances and interacting multiparticle states [73].
3. In his later work, Lukierski's interests are centered upon problems of supersymmetric field theory, namely:
i) In 1978 (in collaboration with V. Rittenberg) he formulated the so called coloured supersymmetry [89] which allows the description of the internal symmetry being the product of flavour and colour symmetries.
ii) In 1979-83 he presented new field theory models describing composite supersymmetric gauge fields [97, 98, 101], composite gravity as well as supergravity [123, 126, 138]. In the papers [97, 101], the models of composite gluons and quarks in supersymmetric chromodynamics were given.
iii) Several papers in 1981-88 were concerned with models of supersymmetric particles and contained the prescription of the first and second quantization of these models. In particular, Lukierski, together with J.A. de Azcarraga, presented in 1981 for the first time the covariant model of a superparticle with nonzero mass [118] and in 1988 the first model of a particle with double supersymmetry (the spinning superparticle) [155, 157, 163, 169, 170]. Both these ideas are quoted and pursued in the world literature up to present time.
4. J. Lukierski with A. Nowicki generalized the supersymmetric formalism of Penrose's twistors to the case of N-extended supersymmetry [149, 156, 17i8] and gave the full classification of the supersymmetric extensions of the Korteweg-de Vries equations [154] (for N = 1,2,3,4; for N = 2 inspection of new terms, for N = 3,4 derivation of new equations).
5. The consecutive work of Lukierski develops a new method of quantization of cohomological complex constraints [157, 162, 175]. It is a variant of BRST (Becchi, Rouet, Stora and Tyutin) formalism describing the generalized quantization of Gupta and Bleuler with holomorphic constraints.
Recently (1990-95) the interest of Lukierski is focused upon the so-called quantum groups. Here are the most important - in my opinion results of Jerzy Lukierski.
i) There was given a system of quantum-deformed (q-deformed) boson as well as fermion creation and annihilation operators covariant under the quantum supergroup. The description of quantum deformations of the supersymmetric oscillator was obtained. This work has been written in collaboration with P. Kulish and M. Chaichian [166, 173].
ii) The first model of quantum four-dimensional Poincaré algebra with the structure of the noncocomutative Hopf algebra was given in 1991. The representation theory of this model was investigated and the first attempts to provide applications were presented (modification of the relativistic kinematics, deformed Klein-Gordon and Dirac equations, corrections to the Lamb shift as well as to the anomalous magnetic moment g--2, etc.). These results were obtained in collaboration with H. Ruegg, A. Nowicki, W. Rühl and V. Tolstoy [176, 179, 183, 195, 202].
iii) Two types of quantum deformations of the four-dimensional conformal algebra were proposed: first one in collaboration with A. Nowicki obtained in 1992 [185] and second very recently in collaboration with P. Minnaert and M. Mozrzymas [223].
iv) There was obtained first in the literature quantum deformation of the four-dimensional Poincaré superalgebra (collaboration with A. Nowicki and J. Sobczyk [196]) as well as four-dimensional Poincaré supergroup (collaboration with P. Kosinski, P. Maslanka and J. Sobczyk [200, 209, 211]).
When looking at this list of recent achievements the results quoted in ii) seem to me to be of particular importance as far as the possible physical implications are concerned. If one replaces the standard classical relativistic symmetries by the corresponding quantum ones (described by the so called quantum Lie algebras and quantum groups) one gets quite new fundamental approach to the problem of space-time structure at small distances. It appears that in this new approach a new fundamental constant with dimension of mass (or length) is added to the well known two constants: $\hbar$ (Planck constant) and c (the light velocity). This constant should control the physical events at small distances where the modification caused by quantum geometry becomes essential. In the theory of deformed relativistic symmetries proposed by Lukierski and his collaborators such a constant appears naturally as a fundamental mass parameter $\kappa$. It should be stressed that from physical point of view the appearance of this parameter $\kappa$ distinguishes favourably the above mentioned theory from other approaches to the quantum deformations of the space-time symmetries. Besides, only for this type of deformation with the fundamental parameter $\kappa$, it is possible to introduce such a deformation of the relativistic symmetry which leaves the three-dimensional nonrelativistic symmetries classical (undeformed).
It should be emphasized that at present some comparisons with the experimental data were performed and the most restrictive estimates imply that the values $\kappa > 10^12$ GeV are admitted. This estimate means that the $\kappa$ -modification of the relativistic symmetries is permitted at the distances $r < 10^{-26}$ cm (a natural assumption seems to be that $\kappa \simeq$ Planck mass $\simeq 10^19$ GeV which corresponds to $r \simeq 10^{-33}$ cm).
The papers of Lukierski, especially the last ones, caused considerable interest among specialists in the whole world. This was testified by the growing number of quotations in the leading international scientific journals as well as by many invitations to significant conferences devoted to the topics pursued by Lukierski. To give an example, Lukierski already in 1978 was invited to the Rochester conference in Tokyo as a chairman of the session named ``New Developments". He was also invited to many plenary talks at the conferences - mostly on applications of group-theoretic and differential-geometric methods in physics. The importance of the scientific results
of Lukierski is also affirmed by the fact that the quantum deformations of relativistic symmetries with a fundamental mass parameter are at present the subject of investigation in many other research centers in Poland Lodz, Warsaw, Cracow) as well as abroad.
In the last 15 years Lukierski led a vast collaboration with well-known scientists abroad. His collaborators were e.g. J.A. de Azcarraga, S. Aoyama, M. Chaichian, W. Heidenreich, A. Isaev, P. Kulish, P. Minnaert, P. Presnajder, V. Rittenberg, H. Ruegg, W. Rühl, V. Tolstoy, P. Vindel and J. van Holten. Nowadays Lukierski plays a leading conceptual role in a considerably big international team working on quantum group problems. He is author or co-author of 223 original contributions published mostly in outstanding scientific journals (Physical Review D, Physics Letters B, Nuclear Physics B, Journal of Mathematical Physics, Annals of Physics, Letters in Mathematical Physics, Nuovo Cimento, Journal of Physics A etc.) as well as in the proceedings of international conferences, schools and workshops (see the attached list of publications).
Jerzy Lukierski was a considerable man as far as the teaching process is concerned. Under his guidance 14 young scientists got their Ph.D. His chair of Elementary Particles and High Energy Physics, comprising mostly his former or actual co-workers and pupils, is the most active scientific group in the Institute of Theoretical Physics of our University. Lukierski was director of three International Winter Schools of Theoretical Physics in Karpacz. The last one, in 1994, was named "Quantum Groups: Formalism and Applications" and was definitely a scientific success.
In 1977-90 Lukierski was scientific deputy director of the Institute of Theoretical Physics of our University. He supervised also for many years the postgraduate studies at our Institute. Since September 1990 he is a director of the Institute.
Approaching sixty, he is a scientist of plenty of energy and new ideas. His moral standards are high. In private life, Jerzy is happily married with Elzbieta, talented artist (26 years younger than him), and a father of two their children: Robert and Ursula. He is always friendly, frank and open towards other people, in particular towards younger ones who need his advice.
I wish him a happy birthday.
Jan Lopuszanski
Institute of Theoretical Physics
University of Wroclaw