Symmetries in Science VIII

Let ℂm,n be the vector space of m×n complex matrices and P(ℂm,n) be the algebra of complex-valued polynomials on ℂm,n. Let GL _m ×GL_n act on P(ℂm,n) by pre-and post-multiplication as follows: $$\left( {{g_1},{g_2}} \right)f\left( x \right) = f\left( {g_1^{ - 1}x{g_2}} \right)$$ where x ∈ ℂm,n, (g₁,g₂) ∈ GL _m ×GL_n,f ∈ P(ℂm,n). We choose a system of coordinates on ℂm,n as follows:$$\left[ {\begin{array}{*{20}{c}}{{x_{11}}}{{x_{12}}} \ldots {{x_{1n}}} \\{{x_{21}}}{{x_{22}}} \ldots {{x_{2n}}} \\\vdots \vdots \cdots \vdots \\{{x_{m1}}}{{x_{m2}}} \cdots {{x_{mn}}}\end{array}} \right]$$

Let F be a field of characteristic 0, let M(n, m) =M(n, m, F) denote the set of n x m matrices over F and let W = W (n, m, F) be the vector space of m-tuples of n x n matrices over F. Let V ⊂ W be a vector space on which a group G ⊂ GL(n, F) acts by simultaneous conjugation. We will denote the polynomial functions on V by P(V) and the G invariants by P(V) G.

Many of us became interested in symmetries of particles and in group theory, a major theme of these series of conferences Symmetries in Science, in the sixties because of the very remarkable symmetry properties of the newly found many fundamental particles at that time. One was very curious about the origin of these symmetries. What have we learned about the origin of these symmetries. What have we learned about these mysterious symmetries after some thirty years? The symmetries were mainly the occurrence of octets and decouplets in the states of mesons baryons and heavy leptons which where attributed to the representations of the internal symmetry group SU(3). Why SU(3), what does it mean? Well, we have not solved it, but pushed the problem a bit further under the carpet, we did not explain the symmetries, nor the group SU(3),but we assigned the same symmetries, or the same quantum numbers to hypothetical particles called quarkes, without really explaining them.

The dynamical group SO(4,2) was first used to gain insight into the H - atom problem. Later it was applied to other systems, such as hadrons, where only scant information exists about the constituents and / or the binding interaction. This is appropriate, since group theoretical treatment of such systems emphasizes global quantum numbers while ignoring internal coordinates that are not accessible to experiment.

The Lie algebra of SU(2) can be extended to the universal enveloping algebra and embedded in a non-commutative, co-commutative Hopf algebra. We demonstrate that this structure, for SU(2), permits a dual structure to be defined, which is again a Hopf algebra of the same type but with a deformed algebra. In the limit of no deformation, this dual Hopf algebra has the Lie algebra of SU(2).

The Grassmann variables in the context of a physical theory are now widely accepted. We can think, for instance, in supergravity or in superstring theories as examples of the importance they have now in contemporary Physics. These already ubiquitous examples are not certainly the only ones. Since the Grassmann variables are used as the classical equivalent of quantum spin, they also appear in the description of systems where the spin plays an important role1. In all these cases, the central idea is based in Dirac’s point of view that we should first try to understand a physical theory, and only then try to quantize it2. To illustrate this point, we quote the work by Crater and Van Alstine3, where they construct two body relativistic wave equations for particles with spin. It is not easy to work at the quantum level with this problem, because one must first try to impose certain conditions of compatibilty on the wave function and in the equations themselves, which restrict the class of available potentials for the problem. Crater and Van Alstine translate the problem to the classical level and then they use the general theory of constraints to analyze it. They find that a supersymmetry condition is needed to impose, in order to obtain consistent equations for the problem. Of course, the final theory is obtained after quantizing their results.

Consider a Lie group G,e.g., the Lorentz, Poincaré,conformal groups,and differential equations 1.1$$ \mathcal{I}\;f = j $$ which are G-invariant. These play a very important role in the description of physical symmetries - recall, e.g., the examples of Dirac, Maxwell equations, (for more examples cf., e.g., [1]). It is important to construct systematically such equations for the setting of quantum groups. Such equations there are expected as q-difference equations. The hope is that these equations will have less singular behaviour than the classical counterparts.

The idea of this contribution goes back to an article published in Symmetry in Science [1]. We proposed there a quantization method for a system S localized on a smooth Riemannian manifold (M,g) and presented preliminary results for the kinematics which were developed and formulated systematically and rigorously in [2,3], with applications in [4]. This approach, the Quantum Sorel Kinematics, is based geometrically on a representation of a pair (B (M), Vect (M)), or equivalently S (M) = (C∞ (M, ℝ), Vect (M)), on some Hilbert space H,with B (M) as the Borel field and Vect(M) as the Lie algebra of smooth vector fields on M,and C∞ (M,ℝ) as the space of smooth functions. For M = ℝ3 the results of the quantization were derived independently in connection with a representation of a certain subgroup of the diffeomorphism group Diff(M) of M in [5]. The pair is a purely kinematical quantity. Sorel sets are generalized positions and vector fields are generalized momenta. To describe a dynamical stuation, S must be furnished with a time dependence. A conventional method to do this is to write an evolution equation or a class of evolution equations. The choice in classical mechanics for point particles is the class of second order (or Newtonian) equations. A construction of a quantum analogue of this class has to be based on the quantization of the kinematics, i.e. on the unitarily inequivalent Quantum Borel Kinematics.

Schrödinger equations for many potentials are solved in terms of the special functions. Almost all of these special functions are the matrix elements of the representations of the Lie groups, with their arguments being the group parameters [1,2]. Even the simplest “special functions” namely the elementary transcendentals are related to the Lie groups, i.e., the one parameter Abelian Lie groups. The connection between the group representations and the special functions explains the mystrious properties of them, such as the recurance relations and addition theorems.

Many “alternative” theories of gravitation have been proposed. Among them, however, the Brans-Dicke theory [1] is certainly one of the best known. This is because of its theoretical simplicity and also because it has been subject to the experimental tests since Dicke himself made an effort by measuring the solar oblateness. Recently the theory seems to be revisited because it offers a sensible approximate model to study the consequences of unification, an ambitious program to unify particle physics and gravitation. In fact in many of the theoretical models of unification scalar fields participate as important ingredients, in addition to other many fields most of which will remain undetected.

Applications of quantum algebras (of su _q (2) as most widespread example) to phenomenological description of rotational spectra of deformed heavy nuclei and diatomic molecules, have appeared a couple of years ago and seem to be encouraging [1–3] (concerning physical applications of quantum groups/algebras in a wider context see ref. [4] and references therein).

Any discrete group W generated by reflections in a space of constant curvature X (discrete reflection group) can be described in terms of its fundamental region P,which is a convex polyhedron whose dihedral angles are proper submultiples of π. By immersing the space X in a linear space E (the ambient space), P extends to a convex polyhedral cone C _p and W extends to a linear group generated by reflections in the faces of C _p (linear Coxeter group).

This paper provides a brief introduction to how unitary representations of diffeomorphism groups can describe certain quantum systems having infinitely many degrees of freedom. It is a partial report of our joint work [1], based on the August 1994 talk by the first author at the Symmetries in Science VIII conference in Bregenz, Austria. We would like to express appreciation to the conference organizers, especially Professor Bruno Gruber, for the opportunity to present our results.

It is well known that, for the simplest possible case of a shell (spin s=1/2, orbital angular momentum I=0), the four shell states transform according to the smallest spin representation (1/2,1/2) of the algebra so(5). In this article it is shown that, within the (space of the) Clifford algebra associated with the fernnion operators, an algebraic shell can be found which is a generalization of the standard shell. The algebraic shell states go over into the standard shell states on the quotient space of the Clifford algebra with respect to a left ideal generated by the identity operator 1 (which corresponds to the standard vacuum state).

In the past ten years two types of coherent state constructions have been used to great advantage to give the matrix representations of group generators and the Wigner coefficients of many higher rank symmetry groups. In both, the irreducible representations of a higher rank group are constructed by an induction process from the irreducible representations of a lower rank subgroup, the so-called core subgroup. In the more widely used first type of vector coherent state construction, [1,2,3], state vectors are mapped onto states of a multidimensional harmonic oscillator through a set of Bargmann variables, z, the so-called “collective” or “orbital” variables, and a set of “intrinsic” or “spin” variables, q _i , which specify the states of the irreducible representates of the core subgroup, the full state vectors being constructed through a “vector-coupling” of the “intrinsic” and “collective” states. This VCS construction has been used for many of the mathematically natural group chains such as U(n) ⊃ U(n-1) x U(1) ⊃ U(n-2) x U(1) ⊃… for which the subgroup chain gives a complete labelling of the state vectors. The VCS construction leads to SU(n) Wigner coefficients which can be expressed in terms of recoupling (Racah or 9j) coefficients of the U(n-1) sugroup, leading to a very simple buildup process for the construction of the Wigner-Racah calculus for the full group.

The three-body problem appears in many branches of physics (Fig.1). Many techniques have been developed to solve the non-relativistic quantum mechanical problem for 3 particles interacting through a two-body or three-body force. These techniques are designed to solve the differential or integro-differential Schrodinger equation. In this article, an alternative formulation of the three-body is given, in terms of Lie algebras. This formulation provides a framework for detailed calculations and, most importantly, allows one to classify all solutions that can be obtained in closed form (exactly solvable problems).

Many years ago Born and Infeld [1] presented a nonlinear electromagnetic field with a non-polynomial action including the so-called universal length. One of the most important characteristics of the Born-Infeld field is found in its static solution which has no infra-red divergence. Many physicists expected that this might be an example of divergence-free field theory. However, no one could succeed to quantize the field, by means of the standard canonical quantization method, because of the complicated nonlinearity. Even the path-integral quantization can hardly be applied to this field, because we cannot easily manipulate such a non-polynomial action. We have to invent a new quantization method if we want to quantize the Born-Infeld field.

The aim of the present paper is to continue the program of extending in the framework of q-deformations the main results of the work in ref. 1 on the SU₂ unit tensor or Wigner operator (the matrix elements of which are coupling coefficients or 3 — jm symbols). A first part of this program was published in the proceedings of Symmetries in Science VI (see ref. 2) where the q-deformed Schwinger algebra was defined and where an algorithm, based on the method of complementary q-deformed algebras, was given for obtaining three-and four-term recursion relations for the Clebsch-Gordan coefficients (CGc’s) of U_q(su2) and U_q(su1,1). The algorithm was fully exploited in ref. 3 where the complementary of three quantum algebras in a q-deformation of the symplectic Lie algebra sp(8, ℝ) was used for producing 32 recursion relations.

Unlike classical physics, modern physics depends heavily on observer’s state of mind or environment. Quantum mechanics depends on how we measure physical quantities, and this issue has not yet been completely settled. In relativity, observers in different Lorentz frames see the same physical system differently. The importance of the observer’s subjective viewpoint was emphasized by Immanuel Kant in his book entitled Kritik der reinen Vernunft whose first and second editions were published in 1781 and 1787 respectively. However, using his own logic, he ended up with a conclusion that there must be an absolute inertial frame, and that we only see the frames dictated by our subjectivity.

Properties of operators of irreducible representations of quantum algebras (q-deformed universal enveloping algebras of Lie algebras) very often differ from these of representation operators for Lie algebras. The main differences are: (a)discrete spectra of operators of representations of finite dimensional representations are mostly non-equidistant(b)closures of unbounded symmetric operators of representations of infinite dimensional irreducible representations are not mostly selfadjoint (in these cases, they have equal deficiency indices and therefore we can construct their selfadjoint extensions)

New generators, relations and subgroups are described for the group Aut(F₃) of automorphisms of the free group F₃. A non-commutative crystallography is described with finite point groups, non-commutative translations, and corresponding space groups. A geometric setting is given in terms of affine transformations.

This work is a review of certain quantum limit theorems that may be viewed as non-commutative versions of the classical central limit theorem. Since various kinds of independences can be introduced in quantum probability, there are many quantum versions of this fundamental result in classical probability. We concentrate here only on various algebraic approaches and even within this scope we do not give a complete survey of the vast literature on the subject.

Following standard methods of crystallography, we investigate the classification of Bravais lattices with respect to vector spaces with a non-Euclidean metric, i.e. we describe sets of transformations of non-compact type that keep an isometry of the Bravais lattice, and apply the method to two and three dimensional spaces.

The contraction of U _q (0(3,2)) |q| = 1 [1, 2] or q real [3]) provided first quantum deformations U _к (P₄) of D = 4 Poincaré algebra P₄≡(M _µv ,P _µ ,) with κ describing the mass-like deformation parameter 1). These so-called ic-deformations are considered in the class of noncommutative and noncocommutative Hopf algebras [4–6]with modified classical coalgebra sector. It should be stressed that the choice of ten generators obtained in [3] is not unique: one can distinguish at least two other bases, with quite interesting properties: i)The bicrossproduct basis, obtained in [7]. In such a basis the quantum algebra U _к P₍₄₎ can be written in the form 2)$${\mathcal{U}_\kappa }\left( {{\mathcal{P}_4}} \right) = \mathcal{U}\left( {O\left( {3,1} \right)} \right)\blacktriangleright \triangleleft T_4^\kappa$$ where—U(0(3,1)) describes the Hopf algebra generated by classical Lorentz, generators, with commutative coproducts$$T_4^\kappa$$describes the K-deformed Hopf algebra of fourmomenta, with commuting generators in algebra sector and K-deformed coalgebra relations.ii)The classical Poincare algebra basis, obtained in [11]3). In such a framework the algebra is a standard Lie algebra, but the coproducts are very complicated noncocommutative expressions.

In this contribution we will review a new approach to some nonlinear dynamical systems which is related to the q-deformation (or other types of deformations) of linear classical and quantum systems considered in [1]. The main idea of this approach is to replace constants like frequency or mass, etc., which are parameters of the linear systems with constants of the motion of the system. This procedure produces from the initial linear system a nonlinear one and as it was demonstrated in [1], [2] the q-oscillator of [3], [4] may be considered as a physical system with a specific nonlinearity which was called q-nonlinearity. In the example of the q-oscillator the constant parameter which was replaced by the constant of the motiom was the frequency, which became dependent on the amplitude of the vibration.

Symmetry presumes the presence of a certain set of representations to be acted upon and transformed among themselves. The phase-space point in classical mechanics and the wavefunction in quantum mechanics are two of the most quoted representations or, more precisely, state-representations in physics. However, representation pointing to a physical object already assumes by itself an implicit operation of getting it while referring to the object in the first place. Any representation of a physical object is intrinsically dynamic in the act of associating the object to its representation. The present implicit dynamics underlying the representation now makes a sharp contrast to the dynamics in terms of representations alone, At issue is how the transformation dynamics of a physical object to its representation would influence the dynamics of representations themselves, especially with regard to its dynamic symmetry. This interference would become most acute when it is focused upon how to conceive of the physical origin of symmetry-breaking acting upon the dynamic symmetry that those representations are supposed to maintain.

A monk saw a turtle walking in the garden of Ta-sui’s monastery and asked his teacher “ All beings cover their bones with flesh and skin. Why does this being cover its flesh and skin with bones?” Ta-sui, the master, took off one of his sandals and covered the turtle with it. ----- The Iron Flute

In 1939 E. Wigner1,2 proposed the induced representation technique when he was obtaining the unitary representation of the Poincaré group. This technique is very useful for elementary particle physics. All elementary particles can be corresponded to the induced representations of the Poincaré group. We have not found any exception.

Recently a lot of attention has been given to generalizations of KdV equations more than one KdV field (see refs.[1–7] and references therein). These integrable coupled equations can play an important role in both physics and mathematics. It is known that there exists a relation between the Poisson bracket structure of integrable systems and the conformal algebras (see refs.[8–13] and references therein). Furtheremore, a connection between integrable systems and conformal field theory has been established after quantizing the integrable systems (see e.g. [14, 15]). Very recently it has also been shown that there is a relation between a matrix KdV hierarchy and a non-linear and non-local Poisson bracket algebra, or the so-called V-algebra [16].

All possible irreducible representations of fundamental algebra for a particle moving on SD are determined by applying the induced representation technique developed by Wigner. It is shown that the theory is automatically equipped with a monopole-like gauge potential. Some topological properties of the gauge potential are also discussed. Examining a relation of our theory with Dirac’s formulation for a constrained system we determine the irreducible representation of the Dirac algebra for a particle constrained on SD.

Although future historians of Physics will remember the first part of the twentieth century for the emergence of Special Relativity and Quantum Mechanics they may well judge that the most fundamental physical discoveries of the century were Einsteins theory of gravitation and the gauge-theory of the fundamental forces. These discoveries completely changed our conception of dynamics whereas Quantum Mechanics and Special Relativity changed only our ideas about kinematics. Furthermore, they showed that Geometry was not only the stage on which physics took place, but was part of the physical drama. The first appreciation of intimate relationship between geometry and physical force came, of course, with the theory of gravitation. In that case the geometry was strictly metrical. The appreciation of the intimate relationship between geometry and the other fundamental forces emerged much more slowly, partly because in those cases the geometry was non-metrical, and non-metrical geometry was itself an innovation. The process took about fifty years altogether, thirty to discover the basic structure of the fundamental forces (a structure which we now know as non-abelian or Yang-Mills gauge theory) and twenty to discover how such a theory should be applied. It is about the first thirty years of this evolution that I wish to speak here.

The low-lying states of baryons are thought to be mainly composed of three-quarks configurations. Although for light (u,d) quarks a construction of the quark-quark interaction is very difficult, it has been suggested that a good fraction of this interaction is of three-body character [1], and that, in addition, it can be well approximated by a hypercentral potential, i.e. an interaction that is invariant under rotations in the six dimensional space spanned by the intrinsic coordinates of the three-body problem. This suggestion arose from a study of the flux tube configurations of Fig. 1 for heavy-quark baryons. However, in general, the fact that QCD is a non-abelian gauge theory leads to gluon-gluon couplings which, in turn, produce three-body forces, as shown in Fig. 2. One is thus led to the conclusion that three-body forces can be a very important part of the quark interactions and this is particularly so for baryons composed of light quarks where the coupling constant α_s is large.

At first, I am very pleased to express my sincere gratitude to Professor Bruno Gruber and all of the people whom we owe the organisation of such a nice, not only from scientific point of view, Symposium. My second remark concerns the motivation of the present work. I am involved since several years in supersymmetries in nuclear physics and during seminars I delivered on our supersymmetry results I have got very often questions from elementary particle physicists: what is the difference between supersymmetry in elementary particle and nuclear physics; whether the Poincaré group is involved in nuclear supersymmetries; is there Clifford Algebra applied to fix the content of supermultiplets; and so on. Hence, I have been forced to l000k more carefully on the elementary particle supersymmetry from the point of view of similarities and differences with nuclear physics. I have been astonished as a new-comer how beauty and dramatic history of the elementary particle supersymmetry is and how many turning points in a supersymmetry development and application have been met. Hence, I would like to pass to you my astonishment connected with several steps in the historical evolution of elementary particle supersymmetries. Then, in the same way I will present the supersymmetry problem in nuclear physics. I will also illustrate the nuclear supersymmetry by examples of our recent results in light nuclei. In the final part I will compare both supersymmetries from mathematical and physical point of view.