Minkowski space Yang-Mills fields from Euclidean self-dual fields

It is examined, if it is possible, to obtain solutions of the SU(2) Yang-Mills field equations in Minkowski space from Euclidean self-dual Yang-Mills fields by method proposed by Bernreuther. It is shown that the conditions, which are imposed on the Euclidean self-dual fields by this method, make every Euclidean self-dual field and the corresponding Minkowski space field obtained from it, equivalent to a pure gauge field, F _ab 0.     

Canonical path-integral quantization of yang-mills field theory with arbitrary external sources

Some modification of source terms is proposed for gauge field theories. In theSU(2) Yang-Mills theory with arbitrary external sources a canonical quantization procedure leads to a Lorentz-invariantS-matrix only when Fermi statistics is imposed on ghost fields. The usual source terms lead to a result that breaks Lorentz invariance and is singular when external chargesJskin0 vanish. The cases of the Abelian scalar electrodynamics and theSU(2) Yang-Mills field with external currents (Jskino=0,Jskini 0) are also discussed.     

Geometry ofSU(2) gauge fields

We studySU(2) Yang-Mills theory onS ³× from the canonical view-point. We use topological and differential geometric techniques, identifying the true configuration space as the base-space of a principal bundle with the gauge-group as structure group.     

Hierarchy of symmetry breakings and neutral currents in anSU(6) grand unification model

In anSU(6) grand unification model with eight quarks and eight leptons belonging to 15-plet and singlet representations, the symmetry is spontaneously broken by the sequenceSU(6)SU(3) ^c ×SU(2)×U(1)×U(1)SU(3) ^c ×U(1). Fror two cases of symmetry breakings the effective weak neutral current coupling constants are compared with experiment. For theSU(3) ^c ×SU(2)×U(1)×U(1)×SU(3) ^c ×U(1) symmetry breaking, the coupling constants reproduce the Weinberg-Salam model with a small correction term. Agreement with the experimental mean values is improved with the correction term. Parity violation in atomic physics is also discussed.     

Yang-Mills fields which are not self-dual

The purpose of this paper is to prove the existence of a new family of non-self-dual finite-energy solutions to the Yang-Mills equations on Euclidean four-space, withSU(2) as a gauge group. The approach is that of equivariant geometry: attention is restricted to a special class of fields, those that satisfy a certain kind of rotational symmetry, for which it is proved that (1) a solution to the Yang-Mills equations exists among them; and (2) no solution to the self-duality equations exists among them. The first assertion is proved by an application of the direct method of the calculus of variations (existence and regularity of minimizers), and the second assertion by studying the symmetry properties of the linearized self-duality equations. The same technique yields a new family of non-self-dual solutions on the complex projective plane.     

Toward finite quantum field theories

The properties that make theN=4 super Yang-Mills theory free from ultraviolet divergences are (i) a universal coupling for gauge and matter interactions, (ii) anomaly-free representations, (iii) no charge renormalization, and (iv) if masses are explicitly introduced into the theory, then these are required to satisfy the mass-squared supertrace sum rule _s=0,1/2(–1)^2s+1(2s+1)M _s ² =0. FiniteN=2 theories are found to satisfy the above criteria. The missing member in this class of field theories are finite field theories consisting ofN=1 superfields. These theories are discussed in the light of the above finiteness properties. In particular, the representations of all simple classical groups satisfying the anomaly-free and no-charge renormalization conditions for finiteN=1 field theories are discussed. A consequence of these restrictions on the allowed representations is that anN=1 finiteSU (5)-based model of strong and electroweak interactions can contain at most five conventional families of quarks and leptons, a constraint almost compatible with the one deduced from cosmological arguments.     

Signals of a Weibel instability in the melting color glass condensate

Based on our work hep-ph/0510121, we discuss further the numerical study of classical SU(2) 3+1-D Yang-Mills equations for matter produced in a high-energy heavy-ion collision. The growth of the amplitude of fluctuations as exp(Γ ) (where g ²μ is a scale arising from the saturation of gluons in the nuclear wave function) is shown to be robust over a wide range of initial amplitudes that violate boost invariance. We argue that this growth is due to a non-Abelian Weibel instability, the scale of which is set by a dynamically generated plasmon mass. We discuss the relation of Γ to the prediction from kinetic theory.     

A spectrum generating algebra for meson resonances

The algebraSO(6,1) is considered as a unification ofSO(6), which is isomorphic toSU(4) SU(3), and the de Sitter algebraSO(4,1). The latter replaces the Poincaré algebra as the algebra of the group of motions of physical space-time. A representation ofSO(6,1) is constructed, which, on restriction toSU(3), decomposes into the direct sum of allSU(3) representations, each occurring just once in the decomposition. The expectation values of the mass-squared operator, when evaluated in the octet, give accurate mass formulae for the octets of 1^– and 2⁺ meson resonances.     

Quaternionic-valued Gravitation in 8D, Grand Unification and Finsler Geometry

A unification model of 4D gravity and SU(3)×SU(2)×U(1) Yang-Mills theory is presented. It is obtained from a Kaluza-Klein compactification of 8D quaternionic gravity on an internal CP ²=SU(3)/U(2) symmetric space. We proceed to explore the nonlinear connection formalism used in Finsler geometry to show how ordinary gravity in D=4+2 dimensions has enough degrees of freedom to encode a 4D gravitational and SU(5) Yang-Mills theory. This occurs when the internal two-dim space is a sphere S ². This is an appealing result because SU(5) is one of the candidate GUT groups. We conclude by discussing how the nonlinear connection formalism of Finsler geometry provides an infinite hierarchical extension of the Standard Model within a six dimensional gravitational theory due to the embedding of SU(3)×SU(2)×U(1)⊂SU(5)⊂SU(∞).     

SU(1, 1) Random Polynomials

We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1, 1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1, 1) random polynomial of degree N are concentrated in a narrow annulus of the order of N ^–1 around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit N. Our results are supported through various numerical simulations. We then extend results of Hannay⁽¹⁾ and Bleher et al. ⁽²⁾ to derive different formulae for correlations between zeros of the SU(1, 1) random analytic functions, by applying the generalized Kac–Rice formula. We express the correlation functions in terms of some Gaussian integrals, which can be evaluated combinatorially as a finite sum over Feynman diagrams or as a supersymmetric integral. Due to the SU(1, 1) symmetry, the correlation functions depend only on the hyperbolic distances between the points on the unit disk, and we obtain an explicit formula for the two point correlation function. It displays quadratic repulsion at small distances and fast decay of correlations at infinity. In an appendix to the paper we evaluate correlations between the outer zeros |z _j |>1 of the SU(1, 1) random polynomial, and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity.     

Charge Conjugation and the B(3) Field: A Reply to Comay

The argument for non-existence of the B ⁽³⁾ field proposed by E. Comay is based on adding radians to the phase of a plane wave. This is trivially incorrect because B ⁽³⁾ is a vacuum component of a C conserving Yang-Mills gauge field theory.     

A spontaneously broken quintet Baryon model for strong interactions

A gauge type model of quantum field theory for strong interactions based on a quinted of observed fields, namely the proton, neutron, , _c and _b baryon fields is proposed. Gauging the resulting global symmetry groupK= SU(3)×¹ U(1)ײ U(1)׳ U(1) for matter fields, one obtains boson-fermion field theory with eleven gauge bosons. The analysis of admissible Higgs sector indicates that the Higgs multiple consists of one adjoint and two fundamental representationsSU(3) and three scalar representations of¹ U(1),² U(1) and³ U(1). The structure of the Higgs sector implies that the original symmetry group extends to the groupK×U(2). Breaking spontaneously the obtained field theory, one converts gauge bosons into the eleven massive vector bosons which can be identified with the observed , K^*, ¯K^*, , , J/ and Y vector mesons. The surviving global symmetry is isomorphic with the symmetry groupSU(2)× ⁰ U(1)× ×¹ U(1)ײ U(1)׳ U(1) corresponding to the isospin, strangeness, baryon number, charm and beauty conservation observed in strong interactions. The surviving Higgs scalars have the same quantum numbers as , K, ¯K, , S, , , and _b mesons. The model gives a newSU(3) classification scheme for baryons without charm and beauty in terms of triplets, sextets and 15-plets. These multiplets can be identified with the observed baryons; the scheme also includes the observed Z₀ and Z₁ baryons (the experimental evidence of which is, nevertheless, still weak). The model predicts the existence and the specific quantum numbers of new mesons and baryons with charm and beauty, and provides a very simple framework for the dibaryon analysis. Since all final physical fields are massive, this model is free from infrared divergences.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

Mass operator in theSU(3)-symmetric strong coupling theory with pseudoscalar mesons

In this paper theSU(3)-symmetric model of a static baryon octet source interacting with pseudoscalar meson octet fields by the coupling of Yukawa type is considered in the strong coupling limit. Using the result derived earlier that the isobaric states form the basis of the unitary irreducible representation of the dynamical group G=T ₂₄ [SU(3) SU(2)], the mass operator is specified by the kinetic part of the Hamiltonian as a particular element of the universal enveloping algebra of the symmetry groupSU(3) SU(2) acting in the space of isobaric states.     

Removable singularities in Yang-Mills fields

We show that a field satisfying the Yang-Mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite. We obtain the result that every Yang-Mills field overR ⁴ with bounded functional (L ² norm) may be obtained from a field onS ⁴=R ⁴{}. Hodge (or Coulomb) gauges are constructed for general small fields in arbitrary dimensions including 4.     

Quantum groupSU(1, 1) ⋊ Z2 and “super-tensor” products

A quantum analogue of the groupSU(1,1)Z ₂—the normalizer ofSU(1, 1) inSL ₂(C)—is introduced and studied. Although there isno correctly defined tensor product in the category of *-representations of the quantum algebraC[SU(1, 1)] _q of regular functions, some categories of *-representations ofC[SU(1, 1)Z ₂] _q turn out to be endowed with a certainZ ₂-graded structure which can be considered as a super-generalization of the monoidal category structure. This quantum effect may be considered as a step to understanding the concept of quantum topological locally compact group.In fact, there seems to be afamily of quantum groupsSU(1, 1)Z ₂ parameterized by unitary characters T ¹ of the fundamental group of the two-dimensional symplectic leaf ofSU(1, 1)/T, whereT is the subgroup of diagonal matrices.It is shown that thequasi-classical analogues of the results of the paper are connected with the decomposition of Schubert cells of the flag manifoldSL ₂(C)_R/B (whereB is the Borel subgroup of upper-triangular matrices) into symplectic leaves.Supported by the Rosenbaum Fellowship.     

ABCD of Instantons

We solve = 2 supersymmetric Yang-Mills theories for an arbitrary classical gauge group, i.e. SU(N), SO(N), Sp(N). In particular, we derive the prepotential of the low-energy effective theory, and the corresponding Seiberg-Witten curves. We manage to do this without resolving singularities of the compactified instanton moduli spaces.To Freeman Dyson, with admirationOn leave of absence from ITEP, Moscow, Russia     

Monopoles,braid groups,and the dirac operator

Using the relation between the space of rational functions on , the space ofSU(2)-monopoles on ³, and the classifying space of the braid group, see [10], we show how the index bundle of the family of real Dirac operators coupled toSU(2)-monopoles can be described using permutation representations of Artin's braid groups. We also show how this implies the existence of a pair consisting of a gauge fieldA and a Higgs field on ³ whose corresponding Dirac equation has an arbitrarily large dimensional space of solutions.The first author was supported by a grant from the NSF     

The semiclassical limit forSU(2) andSO(3) gauge theory on the torus

We prove that forSU(2) andSO(3) quantum gauge theory on a torus, holonomy expectation values with respect to the Yang-Mills measure converge, asT0, to integrals with respect to a symplectic volume measure µ₀ on the moduli space of flat connections on the bundle. These moduli spaces and the symplectic structures are described explicitly.Research supported in part by LEQSF Grant RD-A-08, and NSF Grant DMS 9400961.     

Geometrical Aspects of Skyrmions, Reflection Group, and the Internal Symmetry of Hadrons

The topological aspects of skyrmions are studied and it is shown that hadronscan be viewed as composite states of baby skyrmions when the internal symmetrySU(3) is generated from reflection. It is shown that in an anisotropic space aparticle can move with l = 1/2 with a specific l _x value, and a bosonic constituentmoving with l = 1/2 will appear as a baby skyrmion and a fermionic constituentwill appear as if a spin carrier is attached to a baby skyrmion. The associatedmagnetic field causes a strong statistical attraction which helps to form the boundstate of such constituents. The doublet of such particles having opposite l _x valuesform a conformal spinor when each member behaves as a Cartan semispinor.The conformal reflection then helps us to generate the internal SU(3) symmetry,which splits as SU(3) SU(2) × U(1), giving rise to the hadronic spectra. Thestrong interaction involves a composite cluster in such a bound system whenrearrangement of the constituents takes place preserving the direction vectors,and an elementary constituent can take part in a weak interaction, causing parityviolation. These features help us to consider elementary constituents as knownparticles like leptons.     

Links between neutrino oscillations,leptogenesis, and proton decay within supersymmetric grand unification

Evidence in favor of supersymmetric grand unification including that based on the observed family multiplet-structure, gauge coupling unification, neutrino oscillations, baryogenesis, and certain intriguing features of quark-lepton masses and mixings is noted. It is argued that attempts to understand (a) the tiny neutrino masses (especially Δm ²(v ₂ – v₃)), (b) the baryon asymmetry of the Universe (which seems to need leptogenesis), and (c) the observed features of fermion masses such as the ratiom _b/m_τ, the smallness ofV _cb and the maximality of seem to select out the route to higher unification based on an effective string-unifiedG(224) =SU(2)_L ×SU(2)_R ×SU(2)^c orSO(10)-symmetry that should be operative in 4D, as opposed to other alternatives. A predictiveSO(10)/G(224)-framework possessing supersymmetry is presented that successfully describes the masses and mixings of all fermions including neutrinos. It also accounts for the observed baryon asymmetry of the Universe by utilizing the process of leptogenesis, which is natural to this framework. It is argued that a conservative upper limit on the proton lifetime within thisSO(10)/G(224)-framework, which is so far most successful, is given by x 10³⁴ years. This in turn strongly suggests that an improvement in the current sensitivity by a factor of five to ten (compared to SuperK) ought to reveal proton decay. Implications of this prediction for the next-generation nucleon decay and neutrino-detector are noted.