Relativistic SU(4) and Quaternions

1995

Abstract. A classification of hadrons and their interactions at low energies according to SU(4) allows to identify combinations of the fifteen mesons π, ω and ρ within the spin-isospin decomposition of the regular representation 15. Chirally symmetric SU(2)×SU(2) hadron interactions are then associated with transformations of a subgroup of SU(4). Nucleon and Delta resonance states are represented by a symmetric third rank tensor 20 whose spin-isospin decomposition leads to 4 ⊕ 16 ‘tower states ’ also known from the large-Nc limit of QCD. Towards a relativistic hadron theory, we consider possible generalizations of the stereographic projection S2 → C and the related complex spinorial calculus on the basis of the division algebras with unit element. Such a geometrical framework leads directly to transformations in a quaternionic projective ‘plane ’ and the related symmetry group SL(2,H). In exploiting the Lie algebra isomorphism sl(2,H) ∼ = su∗(4) ∼ = so(5,1), we focus on the Lie alge...

2001

A quaternionic projective theory based on the symmetry group Sl(2,H) allows one to identify various hadron models and many well-known particle transformation laws in its subgroup chains. Identifying the 16-dimensional Dirac algebra {␥ } with Sl(2,H), we use a well-established group-theoretic framework as well as the framework of projective geometry to classify elementary particles and describe their interactions at low energies. It is straightforward to derive Chiral Dynamics and explain the spinorial ('quark') structure of hadrons. Spontaneous symmetry breaking occurs naturally by coset reductions, whereas 'classical' physics is obtained via well-defined limits in terms of a group contraction. The Dirac equation can be identified within a Riemannian globally symmetric space and thus allows one to investigate the fermionic mass as a well-defined parameter. In addition, we suggest an identification of the second quantization scheme and an approach to sum up the perturbation series.

2000

Physical Review, 1965

The spectrum generating algebra for the pairing interaction of quarks is the su4 = so6 Lie algebra independently of the j-shell

2012

The essence of the potential algebra concept [Y. Alhassid, F. Gürsey, F. Yachello. Phys. Rev. Lett. 50 (1983)] is that quantum mechanical free motions of scalar particles on curved surfaces of given isometry algebras can be mapped on 1D Schrödinger equations with particular potentials. As long as the Laplace-Beltrami operator on a curved surface is proportional to one of the Casimir invariants of the isometry algebra, free motion on the surface is described by means of the eigenvalue problem of that very Casimir operator. In effect, the excitation modes considered are classified according to the irreducible representations of the algebra of interest and are characterized by typical degeneracies. In consequence, also the spectra of the equivalent Schrödinger operators are classified according to the same irreducible representations and carry the same typical degeneracies. A subtle point concerns the representation of the algebra elements which may or may not be unitarily equivalent to the standard one generating classical groups like SO(n), SO(p, q), etc. To be specific, any similarity transformations of an algebra that underlies, say, an orthogonal group, always conserve the commutators among the elements, but a non-unitarily transformed algebra must not generate same group. One can then consider the parameters of the non-unitary similarity transformation as group symmetry breaking scales and seek to identify them with physical observables. We here use the potential algebra concept as a guidance in the search for an interaction describing conformal degeneracies. For this purpose we subject the so(4) ⊂ so(2, 4) isometry algebra of the S 3 ball to a particular non-unitary similarity transformation and obtain a deformed isometry copy to S 3 such that free motion on the copy is equivalent to a cotangent perturbed motion on S 3 , and to the 1D Schrödinger operator with the trigonometric Rosen-Morse potential as well. The latter presents itself especially well suited for quark-system studies insofar as its Taylor series decomposition begins with a Cornell-type potential and in accord with lattice QCD predictions. We fit the strength of the cotangent potential to the spectra of the unflavored high-lying mesons and obtain a value compatible with the light dilaton mass. We conclude that while the conformal group symmetry of QCD following from AdS5/CF T4 may be broken by the dilaton mass, it still may be preserved as a symmetry algebra of the potential, thus explaining the observed conformal degeneracies in the unflavored hadron spectra, both baryons and mesons. PACS numbers: 12.39.Jh, 24.85.+p Explorer, your footsteps are the trail, and nothing else: explorer, no trail to follow, advancing trails are blazed. after Antonio Machado To the 70th birthday of Francesco Yachello I. THE POTENTIAL ALGEBRA CONCEPT

HAL (Le Centre pour la Communication Scientifique Directe), 2019

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Advances in Applied Clifford Algebras, 2021

In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with SL(2; H), which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects built from the tensor products of the fundamental representations of SL(2; H) do not carry a representation of the group, due to the non-commutativity of the quaternions. The Lie algebra of the spin group is obtained and its connection with the Lie algebra of SO(5, 1) is presented, providing a physical interpretation for the elements of SL(2; H). Finally, we present a bridge between this quaternionic spinorial formalism for six-dimensional spacetimes and the four-component spinorial formalism over the complex field that comes from the fact that the spin group in six-dimensional Euclidean spaces is given by SU (4).

Revista Mexicana de Fisica

In this paper, we apply Foldy-Wouthuysen transformations to rel- ativistic equation for a free particle of arbitrary spins, which has the symmetry group O(5) associated with it. By noting the isomorphism of O(5) to the symplectic group Sp(4), the mass and spin content of the problem is found using the group reduction Sp(4) U(1) SU(2) where U(1) is associated with the masses while SU(2) is related to the spins. In this paper, we apply Foldy-Wouthuysen transformations to relativistic equation for a free particle of arbitrary spins, which has the symmetry group O(5) associated with it. By noting the isomorphism of O(5) to the symplectic group Sp(4), the mass and spin content of the problem is found using the group reduction Sp(4) U(1) SU(2) where U(1) is associated with the masses while SU(2) is related with the spins. One of the promising approaches to the analysis of relativistic wave equa- tions is connected with Foldy-Wouthuysen (FW) transformations (1). For relativistic equations ...

APS Meeting …, 2001

With a view towards future applications in nuclear physics, a fermion realization of the symplectic sp(4) algebra, isomorphic to so(5), and three q-deformed versions of the algebra are investigated. The deformed representations are based on distinct deformations of the creation and annihilation operators of two kinds of fermions. Three important reduction chains are obtained in the classical limit and in the deformed cases. For the primary reduction SU(2) can be interpreted as either the spin, isospin or angular momentum group of distinct applications. The other reductions describe pairing between fermions of the same kind or of two different types. The reductions all give a complete classification of the basis states. The corresponding u(2) subalgebras of sp(4) are also deformed. They are reducible in the action spaces of sp(4) and decompose into irreducible representations.