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Holomorphic Fueter functions of the position quaternion form a subgroup of Euclidean space‐time diffeomorphisms. An O(4) covariant treatment of such mappings is presented with the quaternionic argument x being replaced by either px or xp involving self‐dual and anti‐self‐dual structures and p denoting an arbitrary Euclidean time direction. An infinite group (the quasiconformal group) is exhibited that admits the conformal group SO(5,1) as a subgroup, in analogy to the two‐dimensional case in which the Mobius group SO(3,1) is a subgroup of the infinite Virasoro group. The ensuing (3+1) covariant decomposition of diffeomorphisms suggests covariant gauges that throw the metric and the stress tensors in standard forms suitable for canonical quantization, leading to ‘‘improved’’ energy‐momentum tensors. Other possible applications to current algebra and gravity will be mentioned.

We derive an effective Hamiltonian of the relativistic quark model. In the limit of zero quark masses, we obtain linear Regge trajectories for mesons. Based on the diquark-antiquark symmetry, we show that the Regge trajectories of baryons and mesons are parallel at high angular moments. We discuss the breaking of the hadronic supersymmetry and obtain a mass relation of mesons and baryons.

Starting from a noncommutative algebra {ital scrA} of the form {ital scrC}{direct product}{ital scrM}, where {ital scrC} is the algebra of smooth functions on space-time and {ital scrM} is the algebra of {ital n}{times}{ital n} Hermitian matrices, we construct an exterior algebra of differential forms over {ital scrA}. We use the one-forms of this algebra to describe Yang-Mills and Higgs fields on a similar footing and construct a Lagrangian from its two-forms. We show how, in the resulting geometrical description, a Higgs potential that leads to spontaneous symmetry breaking arises naturally. We discuss the application of this formalism to the bosonic sectors of the standard electroweak theory and a grand-unified model based on SU(5){direct product}U(1).

In this paper the operator extensions for infinite algebras of residual gauge transformations and space‐time transformations in physical space‐time are discussed on general grounds. Possible general forms of the extensions are presented and their realizations in quantum field theories are discussed. The meaning of nontriviality for the extensions will also be discussed. In particular, cohomology methods are used to calculate the extension of spatial diffeomorphism algebra when chiral fermions are coupled to both an external U(1) gauge field and gravity.

We propose a chiral Lagrangian in which the pions and the vector mesons are treated on equal footing. It gives the usual meson physics and a mass relation for the mesons. We introduce a soliton ansatz for the SU(4) Skyrme model which approximates the more realistic covariant theory. Upon quantization, a rich mass spectrum for the soliton is obtained.

The generators of the subgroups SO(7)L,R and the cosets S7L,R=SO(8)/SO(7)L,R and Σ7L,R=SO(7)L,R/G2 of SO(8) are constructed out of the generators of SO(8) using octonionic structure tensors, and the representations of these generators are given as products of octonions.

Octonionic extensions of Quantum Mechanics associated with finite exceptional Hilbert spaces are summarized. An interpretation of such spaces as the finite space of internal symmetries allows the introduction of color and flavor degrees of free dom for fundamental particles, color arising from the automorphism group of octonions. As an example the space admitting E symmetry is discussed. The projection operators for various fermion states are constructed and shown to belong to a larger ternary algebra of hermitian operators. A possible application is to the grand unified gauge theory of leptons and quarks based on E6. Another possibility involves the construction of an exceptional superspace by means of octonionic Grassman numbers.

If the exceptional observables introduced by Jordan, von Neumann and Wigner are identified with charge states corresponding to internal degrees of freedom of elementary particles, then one is led to the classification of quarks and leptons by means of exceptional groups. It is shown that the groups of the E series are likely candidates and a gauge field theory based on E7 is given as an example. The hierarchy of symmetry breaking is linked to the hierarchy of stability groups for geometries that are associated with the exceptional groups.

It is shown that self-dual theories generalize to four dimensions both the conformal and analytic aspects of two-dimensional conformal field theories. In the harmonic space language there appear several ways to extend complex analyticity (natural in two dimensions) to quaternionic analyticity (natural in four dimensions). To be analytic, conformal transformations should be realized on $CP^3$, which appears as the coset of the complexified conformal group modulo its maximal parabolic subgroup. In this language one visualizes the twistor correspondence of Penrose and Ward and consistently formulates the analyticity of Fueter. It is shown that self-dual theories generalize to four dimensions both the conformal and analytic aspects of two-dimensional conformal field theories. In the harmonic space language there appear several ways to extend complex analyticity (natural in two dimensions) to quaternionic analyticity (natural in four dimensions). To be analytic, conformal transformations should be realized on $CP~3$, which appears as the coset of the complexified conformal group modulo its maximal parabolic subgroup. In this language one visualizes the twistor correspondence of Penrose and Ward and consistently formulates the analyticity of Fueter.

An approach to a unified treatment of Yang-Mills and Higgs fields was proposed previously starting from a noncommutative algebra constructed from the algebra of smooth functions on space-time and the algebra of Hermitian matrices. The formalism was further applied to the bosonic sector of the standard electroweak theory. Here, in order to incorporate fermions into this framework in a natural way, we extend the formalism to include supersymmetry. We find, however, that the scenario changes drastically in the presence of supersymmetry and the conclusions are substantially different. When applied to the standard model, this yields the supersymmetrized bosonic sector of the theory, but the inclusion of quarks and leptons requires additional superfields as in the conventional supersymmetric models.

Both bound and scattering states of the P\"oschl-Teller potential are shown to be connected with unitary representations of certain groups. A family of periodic potentials, and their associated transfer matrices and band structure, can also be obtained from group theory and reduce to the above potential when the real period approaches infinity. These results suggest that the algebraic approach used for treating bound-state problems can be extended to scattering and band-structure problems.

We exhibit two octonionic extensions of the Kalb-Ramond type fields with rank two and four in eigth dimensions. By analogy to the d = 4 (anti-) self-duality of the SU(2) ≈ S 3 quaternionic gauge field we consider the respective d = 8 (anti-) self-duality equations for these nonlinear, S 7 -valued antisymmetric fields. By way of an octonionic 't Hooft ansatz these equations reduce to the same generalized Fueter-Cauchy-Riemann equations over S 8 . Explicit (9 n + 8) parameter S 7 → S 7 mapping solutions, n being a winding number, are found in terms of holomorphic functions of the spacetime octonion. An infinite number of local continuity equations results.

It is shown that a Lorentz covariant coordinate system can be chosen in the case of the Kerr–Schild geometry which leads to the vanishing of the pseudo energy–momentum tensor and hence to the linearity of the Einstein equations. The retarded time and the retarded distance are introduced and the Lienard–Wiechert potentials are generalized to gravitation in the case of world‐line singularities to derive solutions of the type of Bonnor and Vaidya. An accelerated version of the de Sitter metric is also obtained. Because of the linearity, complex translations can be performed on these solutions, resulting in a special relativistic version of the Trautman–Newman technique and Lorentz covariant solutions for spinning systems can be derived, including a new anisotropic interior metric that matches to the Kerr metric on an oblate spheroid.

A comparative and systematic study is made of 2-dimensional CP(n) σ-models and new 4-dimensional HP(n) σ-models and their respective embedded U(1) and Sp(1) holonomic gauge field structures. The central theme is complex versus quaternionic analyticity. A unified formulation is achieved by way of Cartan's method of moving frames adapted to the hypercomplex geometries of the harmonic symmetric spaces CP(n) ≈ SU(n + 1)SU(n) × U(1) and HP(n) ≈ Sp(n + 1)Sp(n) × Sp(1) respectively. Elements of complex Kahler manifolds are applied to a detailed analysis of the CP(n) σ-model and its instanton sector. Generalization to any Kahlerian σ-model is manifest. On the basis of Cauchy-Riemann analyticity, Kahlerian models are shown to have an infinite number of local continuity equations. In a parallel manner, new 4-dimensional conformally invariant HP(n) σ-models are constructed. Focus is on the latter's hidden local gauge invariance in their holonomy group Sp(n) × Sp(1) which allows a natural embedding of the Sp(1) ≈ SU(2) pure Yang-Mills theory. The associated quaternionic structure is discussed in light of both quaternionic quantum mechanics and Kahlerian geometry. In this chiral setting, the SU(2) Yang-Mills duality equations are cast into quaternionic Cauchy-Riemann equations over S4 ≈ HP(1), the conformal spacetime. In analogy to the CP(n) case, their rational solutions are the most general (8n − 3) parameter instantons where the associated algebraic nonlinear equations of the type of Atiyah, Drinfeld, Hitchin, and Manin are now expressed in a new conformally invariant form. Geometrically, the SU(2) instantons solve the Frenet-Serret equations for quaternionic holomorphic curves; they are conformal maps from HP(1) into HP(n) with n their second Chern index. Fueter's quaternionic analysis is presented, then applied: Fueter functions are particularly suited for the solutions of 't Hooft, of Jackiw, Nohl and Rebbi, and of Witten and Peng, as well as the self-dual finite action per unit time solution of Bogomol'nyi, Prasad and Sommerfield. Generalizing the latter, a new solution with unit Chern index and finite action per unit spacetime cell is found. It is expressed in terms of the quaternionic fourfold quasi-periodic Weierstrass Zeta function. Finally the essence of our method is revealed in terms of universal connections over Stiefel bundles; generalization to real, complex and quaternionic classifying Grassmanian σ-models with their embedded SO(m), SU(m) and Sp(m) gauge fields is outlined in terms of gauge invariant projector valued chiral fields. Other outstanding problems are briefly discussed.

It is shown that a system of first-class bosonic constraints obeying a Lie algebra has associated with it a natural superalgebra. BRST quantization arises as a non-linear representation of this superalgebra. Two distinct superalgebras are explicitly constructed and their associated BRST quantizations presented. The first BRST quantization is the canonical one with the BRST charge a grassmannian scalar. The second is new — the BRST charge is a grassmannian spinor transforming in the fundamental representation of the appropriate superalgebra. Generalizations are briefly discussed.

Miyazawa's supersymmetric generalization of the approximateSU 6 symmetry of hadrons, namelyU 6/21, is shown to arise within the frame of the standard theory of quarks interacting through gluons. The supersymmetry is found to hold in the same approximation for whichSU 6 is a good symmetry, provided a diquark structure emerges through an effective string interaction. The spin independence of the confining force of the strong-coupling regime then explains the near parallelism of the leading Regge trajectories for baryons and mesons. Symmetry breaking is due to mass differences of the constituents and spin-dependent forces caused by gluon exchange. TheU 6/21 algebra is also shown to be embedded in a larger octonionic algebra which puts mesons and baryons, exotic mesons, quarks and diquarks in the same multiplet. The experimental evidence for the existence of qq and $$qq\overline {qq} $$ systems is briefly discussed.

It is shown that Miyazawa’sSU(6/21) superalgebra for hadrons can be applied to a bilocal quark-antiquark (q\(\bar q\)) or quark-diquark (q-D) system resulting in multiplets that include thel=1 mesons in the (35+1) representation ofSU(6) and thel=1 baryons (70−), in addition to thel=0 mesons (35−+1−) andl=0 baryons (56+). Alternatively, a nonsimple algebra that brings together the (35−+1−) mesons and 56+ baryons is also proposed. This minimal scheme does not involve the D-\(\bar D\) exotic mesons in the lowest hadronic supermultiplet. In both schemes color is incorporated through the algebra of split octonions.

Starting from the fact that the d=4 Euclidean flat spacetime is conformally related to the Kahler manifold H2×S2, we show the Euclidean Schwarzschild metric to be conformally related to another Kahler manifold M2×S2 with M2 being conformal to H2 in two dimensions. Both metrics which are conformally Kahlerian, are form-invariant under the infinite parameter Fueter group, the Euclidean counterpart of Milne's group of clock regraduation. The associated Einstein's equations translate into Fueter's quaternionic analyticity. The latter leads to an infinite number of local continuity equations.

In an explicit, unified, and covariant formulation, we study and generalize the exceptional vector products in R8 of Zvengrowski, Gray, and Kleinfeld. We derive the associated general quadratic, cubic, and quartic G2 invariant algebraic identities and uncover an octonionic counterpart to the d=4 quaternionic duality. When restricted to seven dimensions the latter is an algebraic statement of absolute parallelism on S7. We further link up with the Ogievetski–Tzeitlin vector product and obtain explicit tensor forms of the SO(7) and G2 structure constants. SO(8) transformations related by Triality are characterized by means of invariant tensors. Possible physical applications are discussed.

We demonstrate that the BRST charge operator is the generator of a nonlinear (fractional linear) transformation within the supergroup obtained by exponentiating the superalgebra whose even part is a set of first class bosonic constraints forming a Lie algebra and whose odd part is composed of the conjugate ghost operators and the BRST charge. Two alternative constructions are given. In the first the charge is the conventional singlet and in the second it is a spinor. The two constructions correspond to embedding the bosonic Lie algebra in two different superalgebras.

The power of the Dirac algebra is illustrated through the Kahler correspondence between a pair of Dirac spinors and a 16-component bosonic field. The SO(5, 1) group acts on both the fermion and boson fields, leading to a supersymmetric equation of the Dirac type involving all these fields.

In a conformally invariant scalar field theory on the three-sphere, a scalar-spinor theory with Yukawa interaction, Higgs Mechanism and fixed couplings is shown to arise from the curved geometry. This model admits a threefold description and an exact one-kink solution. Implications are discussed.

We show that both bound and scattering states of a certain class of potentials are related to the unitary representations of certain groups. In this class, several potentials of practical interest, such as the Morse and Poschl-Teller potentials, are included. The fact that not only bound states but also scattering states are connected with group representations suggests that an algebraic treatment of scattering problems similar to that of bound state problems may be possible.

We use the potential group SO(2, 2) to realize a class of solvable potentials. The scattering matrices can be obtained by purely algebraic techniques.

A universal gauge theory based on E6 is suggested as a model for the unification of strong, electromagnetic and weak interactions. Left and right components of six quarks (3 light, 3 charmed) and nine leptons (including heavy charged and neutral leptons) are fifted into two 27-dimensional representations of E6. Two different assignments are proposed. The Weinberg angle and the R ratio are derived and the stability of the proton is discussed.

The discrete Jordan Algebras of 1×1, 2×2 and 3×3 hermitian matrices over integer elements of the four division algebras are considered. They are transformed under discrete subgroups of groups associated with the magic square. Points corresponding to a discrete Jordan matrix belong to a lattice generated by Weyl reflections that are expressed by means of Jacobson’s triple product. Special cases include the O(32), E8×E8 and E10 lattices that occur in superstring theories.

We show that the calculation of the S-matrix associated with a class of solvable potentials with SU(1, 1) dynamic symmetry can be formulated in purely algebraic terms. This formulation makes use of the Euclidean group in two dimensions, E(2), and is amenable to generalization to a larger number of dimensions.

Spinor representation of the Schwarzschild metric is given. It is shown how the Trautman-Newman complex translation leads to the Kerr metric in this representation. Some new solutions are found when λμ appearing in the Kerr-Schild metric is a constant null vector. In this case, it is observed that the Kerr-Schild metric is the solution of the Einstein-Maxwell field equations with a null fluid. It is also found that, for some special cases, this metric is a solution for the electrovacuum gravitational-field equations with a cosmological constant.

The Super Poincaré Groups in d = 3, 4, 6 and 10 are represented by graded matrices over the four division algebras. It is shown that classical superstring actions can be written in the same dimensions in terms of the supergroup elements. Possible applications are discussed.

A group theoretical approach to the one-dimensional Morse oscillator, including both bound and scattering states, is presented. It is shown that the group describing the scattering states, U(1,1), can be obtained from that describing the bound states. U(2), by analytic continuation. The inclusion of the continuum allows one to treat algebraically dissociation and scattering in a Morse potential.

An algebraic technique, useful in studying relativistic scattering of a Dirac particle in a certain class of external fields, is presented. As an example, this technique is applied to the algebraic determination of the S-matrix for Coulomb-Dirac scattering. The technique requires the introduction of linear invariant operators or linear invariant coset operators (LCO).

We study the graded Lie groups corresponding to the graded Lie algebras SU(2,2/1) and OSp(1/4). General finite group transformations are parametrized, and nonlinear representations are obtained on coset spaces. Jordan and traceless algebras are constructed which admit these groups as automorphism groups.

The formalisms of spontaneous symmetry breaking and of the theory of nonlinear Lagrangians are reviewed, with emphasis on the relationships between the two. The former is applied to the Weinberg-Salam model, in which case the physical fields and their explicit transformation laws under finite group action are exhibited. The latter is illustrated through the example of the graded Lie group OSp(1/4). Locally covariant fields and derivatives are then constructed, and invariant terms are written down which contain the Lagrangian of supergravity.

Four-dimensional second-order supersymmetric equations involving an arbitrary function of an unconstrained chiral superfield are proposed. In three dimensions a new invariant operator provides the generalization of analogous two-dimensional equations that are of first order in the fermion field.

In an octonionic realization of the Cartan-Schouten parallelism on S 7 we exhibit the algebraic nature and Hopf topology of Englert's solutions for spontaneously compactified d =11 supergravity. The S 7 torsion and its dual, the index 4 gauge field are given locally by the Cayley structure constants and the associator of the imaginary octonion units, respectively.

Compact multi S 4 gravitational instantons (positive definite metric solutions to Einstein's equations with a Λ term) are found in terms of quaternionic harmonic mappings x n : S 4 → S 4 . The manifold is the euclidean de Sitter space S 4 ≃ HP(1) with Euler-Poincare index I E = n and Pontryagin index I P = 0. The metric can be alternatively seen as the induced metric of a Kahler HP(1) chiral field space, these same solutions then correspond to all multi chiral instantons in a general covariant quaternionic HP(1) ≃ SO(5)/SO(4) ≃ Sp(2)/Sp(1) × Sp(1) nonlinear σ-model of de Alfaro et al. They are the quaternionic counterparts of the harmonic instanton maps of the 2-dimensional CP(1) ≃ SO(3)/SO(2) ≃ SU(2)/U(1) σ-model.

Particle assignments in a unified theory proposed previously are re-examined in both topless and standard versions. Spontaneous symmetry breaking via Higgs fields transforming as 27, 78 and 351 is examined and possible distributions of v.e.v.'s are discussed in both cases. The generation of a 351 effective Higgs field in a two-loop diagram that gives mass to the right-handed neutrino fields is shown to be compatible with the standard model.

The relevance of the representation theory of the conformal group to quantum field theory is illustrated in two dimensions by the Thirring model. Space-time position operators and their complex extensions are defined as operator-valued functions of the generators. These complex position variables coincide with the Gel'fand-Naimark labels and can be interpreted as labels for nonorthogonal coherent states. A Hilbert-space metric then becomes necessary. It is given by the matrix elements of the metric operator $G$ and is nontrivial for the nonanalytic supplementary series and the analytic representations. In this case ${G}^{\ensuremath{-}1}$ gives the two-point function for the Thirring model. For nonanalytic representations only weak (infinitesimal) conformal invariance holds for interacting fields if causal and spectral properties are imposed, while those properties become compatible with strong (global) conformal invariance in the case of analytic representations which lead either to free fields or to interacting fields with a quantized value of the coupling constant.

The formulation of weak interactions by means of a current interacting with itself is generalized to the case in which T = 32 currents also exist, assuming the ΔT = 12 rule to be valid. Furthermore, generalizing the Feynman-Gell-Mann theory, it is postulated that all the currents entering the weak interactions are conserved in the limit of meson masses and baryon mass differences being neglected. This restriction limits the isotopic spin of the currents to T = 12, 1, and 32 if S = 2 currents are excluded. Another consequence of the same hypothesis is that parity mixtures are allowed for the T = 12 and T = 1 currents, but not for the T = 32 current. The five types of currents thus obtained (V and A currents with T = 12, 1 and only V or A with T = 32) can be combined into a 4 × 4 matrix current |Jχ. It is shown that a weak Lagrangian proportional to the trace of |Jλ†|Jλ satisfies the ΔT = 12 rule. In the absence of neutral lepton currents the charged lepton current is necessarily incorporated in a rather unsymmetrical manner, but in conformity with the ΔT = 12 rule in strangeness changing leptonic decays. The scheme is applied to the decay of Σ hyperons. In the pionic decay of Σ+, and in the lowest order in strong interactions, parity is conserved in the channel Σ+ → n + π+ and violated in the channel Σ+ → ϱ + π0 if all the five types of current are present. The phenomenological Lagrangian of Bludman for Σ decay is obtained as a direct interaction from the terms of the weak Lagrangian. Finally, a model of a baryonmeson nonlinear interaction generalizing a previous model is given in which all the five types of conserved currents are shown to exist.

A convenient description of hadron states is presented by means of minimum-uncertainty wave packets in transverse internal position or momentum operators which can be defined in the infinite-momentum frame (IMF) basis for the Poincare group. The center of mass of the hadron is described either by its momentum or its IMF position operator, while its internal structure is fixed by taking coherent states symmetrical in the internal co-ordinates. The resulting physical picture of hadrons as extended objects with a stable structure agrees with Yang’s model of droplets with optimum universal size. The reduction of products of states in the infinite-momentum frame with respect to our basis introduces Clebsch-Gordan coefficients which asymptotically depend only on certain scaling variables, thus allowing us to trace their emergence to a general framework. These are precisely the scaling variables connected with different high-energy regions where limiting behavior patterns for hadronic cross-sections, like diffraction, Bjorken scaling, Feynman scaling and limiting fragmentation are known to occur.

It is argued that Einstein's Theory of General Relativity as it stands incorporates Mach's Principle. The boundary conditions for Machian solutions are stated in a coordinate system in which the cosmological background is described by a conformally flat metric. The metric tensor g μν is then written as a product of the scalar density ϕ 2 and a tensor density γ μν with unit determinant. In the coordinate system that has been so chosen ϕ describes the cosmological structure, while γ μν refers to gravitational phenomena. This becomes clear when Einstein's fundamental equations are rewritten in terms of ϕ and γ μν . Then κϕ −1 is seen to play the role of the gravitational constant instead of κ in the weak field approximation. The quantity κϕ −1 can be expressed in terms of the radius and the total mass of the universe and the sign of the forces between inhomogeneities of the metric is determined by the requirements of Mach's principle. The forces which derive from ϕ are found to be repulsive for the cosmological background, leading to the expansion of the universe, while attractive gravitational forces arise from the deviations of γ μν from the Minkowski metric. Various statements associated with Mach's Principle are discussed in the light of this reformulation of Einstein's Theory.

A quark model with a core that was previously suggested is used to define a trilocal field to describe mesons. The field is taken to satisfy a second-order equation which is Lorentz invariant and invariant under the linear group acting on the two internal co-ordinates. The local fields occurring in the expansion of the trilocal field then describe mesons with spinJ and multiplicityJ on the leadingρ-A2 linear trajectory. This result shows that such a scheme can accommodate the doubling of the A2-meson and predicts tripling for the R-meson.

A simple theory of gravitation is formulated in conformal Riemannian space-time. The metric is determined by a scalar function which satisfies a linear equation. A conclusion in favour of Einstein's general tensor theory is drawn from a discussion of the corrections to the Newtonian theory for purely gravitational phenomena. Finally the theory is applied to the cosmological problem and especially to the possibility of a steady-state universe. The velocity-distance law is shown to be compatible with a constant uniform distribution of matter without the need of artificial assumptions.

The non-linear conform-invariant spinor wave equation proposed in an earlier paper is given a simple geometrical interpretation within the frame of general relativity. At each point of a conformal space-time manifold, an orthogonal frame of reference is defined, this frame being associated with a 4-spinorψ. It is shown that if the contracted torsion tensor ια relative to the frame vanishes (as in the case of a co-ordinate transformation) and the pseudo-vectorLα derived from the same torsion tensor remains constant and space-like, then the associated 4-spinorψ satisfies just the above mentioned wave equation which, in turn, determines the orthogonal frame of reference at every space-time point to within a conformal transformation of the co-ordinate system. In particular, if the pseudo-vectorLα also vanishes one obtains Dirac’s equation for a particle of rest mass zero. In a discussion of the physical meaning of the orthogonal frame of reference, the time-like basis vector and one of the space-like basis vectors are respectively related to the current density and the spin density 4-vectors associated with the wave equation.