An action principle for the Neveu-Schwarz-Ramond string and other systems using supernumerary variables

An action principle which gives rise to the equations of motion and boundary conditions for the free relativistic string with fermionic degrees of freedom is presented. With the aid of extra variables, some of which are Grassmann functions, all the gauge generators are obtained as secondary constraints. The consistency of the system is demonstrated using a generalised Poisson bracket operation. The theory is quantised with Dirac brackets and the fermionic fields become elements of a Clifford algebra. The methods are also used to formulate the theory of the Klein-Gordon and Dirac point particles and the relativistic string and membrane without intrinsic spin. Under certain circumstances we show that the supernumerary variables may be removed entirely from the original Lagrangian.     

Higgs fields and the determination of the Weinberg angle

SU(2) × U(1) gauge theories, in which the Higgs fields transform as doublets under SU(2) are interpreted as pure Yang-Mills theories in six dimensions, the components of the gauge potentials in the extra dimensions playing the role of the Higgs' fields. Two consistent theories are discovered: one in which SU(2) × U(1) is embedded in SU(3) and the vector bosons remain massless - and another where SU(2) × U(1) is embedded in the graded Lie algebra SU(2|1), the symmetry is spontaneously broken in a natural fashion and the theory is equivalent to that of Weinberg and Salam, with a specific value 30° for the Weinberg angle and a prediction of the Higgs' mass.     

Locally gauge-invariant formulation of parastatistics

A locally gauge-invariant formulation of parastatistics, which is equivalent to a Yang-Mills gauge theory, is given, using a complex Clifford algebra (case of SU(N)) or a real Clifford algebra (case of SO(N)). In particular, for the SU(3) case, the gauged theory of para-Fermi quarks is equivalent to quantum chromodynamics.     

The soliton supermultiplet in 4 + 1 dimensions

We consider the SO(4) = SU(2) ? USp(2) Clifford algebra, obtained by the supersymmetry algebra for the N = 2 supersymmetric Yang-Mills theory in 4+1 dimensions, which, in the phase of unbroken gauge symmetry, has a topological charge as central charge. We find that, even if the Higgs mechanism is absent, the massive soliton supermultiplet contains the same number of states as the massless supermultiplet of elementary particles.     

Signature Change and Clifford Algebras

Given the real Clifford algebra of a quadratic space with a given signature, we define a new product in this structure such that it simulates the Clifford product of a quadratic space with another signature different from the original one. Among the possible applications of this new product, we use it in order to write the Minkowskian Dirac equation over the Euclidean spacetime and to define a new duality operation in terms of which one can find self-dual and anti-self-dual solutions of gauge fields over Minkowski spacetime analogous to the ones over Euclidean spacetime and without needing to complexify the original real algebra.     

Augmented superfield approach to unique nilpotent symmetries for complex scalar fields in QED

The derivation of the exact and unique nilpotent Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetries for the matter fields, present in any arbitrary interacting gauge theory, has been a long-standing problem in the framework of the superfield approach to the BRST formalism. These nilpotent symmetry transformations are deduced for the four (3+1)-dimensional (4D) complex scalar fields, coupled to the U(1) gauge field, in the framework of an augmented superfield formalism. This interacting gauge theory (i.e. QED) is considered on a six (4,2)-dimensional supermanifold parametrized by four even spacetime coordinates and a couple of odd elements of the Grassmann algebra. In addition to the horizontality condition (that is responsible for the derivation of the exact nilpotent symmetries for the gauge field and the (anti-)ghost fields), a new restriction on the supermanifold, owing its origin to the (super) covariant derivatives, has been invoked for the derivation of the exact nilpotent symmetry transformations for the matter fields. The geometrical interpretations for all the above nilpotent symmetries are discussed, too. PACS 11.15.-q, 12.20.-m, 03.70.+k     

Infinite-dimensional gauge groups and special nonlinear gravitons

A gauge theory in flat space—time, in which the gauge algebra is the (infinite-dimensional) algebra of vector fields on a surface, determines a curved space—time metric. This note deals with some completely integrable examples, concentrating on the N → ∞ limit of the Euler—Arnol'd equations [geodesics on SO(N)]. In this case, the metric turns out to be flat, which points the way to a coordinate transformation that solves the original equations.     

Soft gauge algebras

Gauge transformations whose algebra closes only modulo field dependent terms (soft gauge algebras) are studied in detail. The results are explicitly applied to a supersymmetric gauge theory, to gravity and to conformal gravity, all seen as gauge theories overx-space; the obvious applications to supergravity are pointed out. A consistency requirement for the gauge transformations of those fields which appear in the algebra is seen to rule out “local translations” as independent gauge transformations.     

Duality in Off-Shell Electromagnetism

In this paper, we examine the Dirac monopole in the framework of Off-Shell Electromagnetism, the five-dimensional U(1) gauge theory associated with Stueckelberg–Schrodinger relativistic quantum theory. After reviewing the Dirac model in four dimensions, we show that the structure of the five-dimensional theory prevents a natural generaliza tion of the Dirac monopole, since the theory is not symmetric under duality transforma tions. It is shown that the duality symmetry can be restored by generalizing the electromagnetic field strength to an element of a Clifford algebra. Nevertheless, the generalized framework does not permit us to recover the phenomenological (or conventional) absence of magnetic monopoles.     

U(4) spin gauge theory over a Riemann-Cartan space-time

A spin gauge theory based on the groupU(4) is investigated in a general relativistic context including the possibility of nonzero torsion. The language of Clifford bundles over a space-time with metric and metric compatible torsion is used as a convenient tool for the study of fields defined on space-time possessing Clifford multiplication properties. A Dirac-type representation is investigated in detail and the geometric implications for spin gauge theory are pointed out.     

Fermi-Bose hypersymmetry

A new algebra, combining supersymmetry and internal symmetry, is presented. A massless vector hypermultiplet contains a vector, an isodoublet of left-handed Dirac spinors, and a complex scalar. These can be used as generalized gauge fields. Abelian as well as non-Abelian gauge theories are studied, and the Higgs mechanism is extended in a hypersymmetric way. We present, also, a (mom-realistic) SU(2)× U(1) model; gauge invariance and hypersymmetry are spontaneously broken; two Goldstone spinors appear. Hypersymmetry allows one to define “electronic” and “muonic” numbers, and suggests that a weakly interacting scalar particle ωγ is associated with the photon and the two neutrinos.     

Reply to da Rocha and Rodrigues' comments on the orientation congruent algebra and twisted forms in electrodynamics

The recent claim by da Rocha and Rodrigues that the nonassociative orientation congruent algebra (???? algebra) and native Clifford algebra are incompatible with the Clifford bundle approach is false. The new native Clifford bundle approach, in fact, subsumes the ordinary Clifford bundle one. Associativity is an unnecessarily too strong a requirement for physical applications. Consequently, we obtain a new principle of nonassociative irrelevance for physically meaningful formulas. In addition, the adoption of formalisms that respect the native representation of twisted (or odd) objects and physical quantities is required for the advancement of mathematics, physics, and engineering because they allow equations to be written in sign‐invariant form. This perspective simplifies the analysis of, resolves questions about, and ends needless controversies over the signs, orientations, and parities of physical quantities.     

Weil algebra structure and geometrical meaning of BRST transformation in topological quantum field theory

The Weil algebra structure of the BRST transformation of topological quantum field theory is investigated. This structure appears in the gauge and ghost fields sector and is common to both topological quantum field theory and BRS gauge fixed non-abelian gauge theory. By the Weil algebra structure, we can derive the descent equations of topological quantum field theory which generate the Donaldson polynomials. The algebraic structure also reveals the geometrical meaning of the ghost fields ψ and ? in topological quantum field theory as the components of the total curvature.     

The construction of self-dual solutions to SU(2) gauge theory

Ignoring the problem of sources and singularities, explicit expressions are constructed for the ansätze of Atiyah and Ward. These take an especially simple form in theR gauge of Yang. Some non-linear transformation properties of the self-duality equations in this gauge provide an inductive proof of the ansätze. There is a six-parameter family of these Bäcklund transformations. They take real SU(2) gauge fields into real SU(1, 1) gauge fields and vice versa.On leave of absence from Department of Mathematics, University of Durham, U.K.     

Equations of motion of interacting massless fields of all spins as a free differential algebra

It is argued that the equations of motion of interacting massless fields of all spins s=0,1,…,∞ can naturally be formulated in terms of a free differential algebra (FDA) constructed from one-forms and zero-forms that belong both to the adjoint representation of the infinite-dimensional superalgebra of higher spins and auxiliary fields proposed previously. This FDA is found explicitly in the first non-trivial order in the zero-forms. Various properties of the proposed FDA are discussed including the ways for incorporating internal (Yang-Mills) gauge symmetries via associative algebras.     

Twisted Gauge Theories

Gauge theories on a space-time that is deformed by the Moyal–Weyl product are constructed by twisting the coproduct for gauge transformations. This way a deformed Leibniz rule is obtained, which is used to construct gauge invariant quantities. The connection will be enveloping algebra valued in a particular representation of the Lie algebra. This gives rise to additional fields, which couple only weakly via the deformation parameter θ and reduce in the commutative limit to free fields. Consistent field equations that lead to conservation laws are derived and some properties of such theories are discussed.     

Clifford algebra structure and Einstein's equations

We point out that a Clifford algebra representation for the Riemannian curvature leads to an equation for gravity similar to the Yang-Mills equation, in a gauge model for gravity with the Lorentz group. Einstein's equation of general relativity emerges as a natural solution in this approach.     

Weyl group, <Emphasis Type="Italic">CP</Emphasis> and the kink-like field configurations in the effective <Emphasis Type="Italic">SU</Emphasis>(3) gauge theory

Effective Lagrangian for Yang-Mills gauge fields invariant under the standard space-time and local gauge SU(3) transformations is considered. It is demonstrated that a set of twelve degenerated minima exists as soon as a nonzero gluon condensate is postulated. The minima are connected to each other by the parity transformations and Weyl group transformations associated with the color su(3) algebra. The presence of degenerated discrete minima in the effective potential leads to the solutions of the effective Euclidean equations of motion in the form of the kink-like gauge field configurations interpolating between different minima. Spectrum of charged scalar field in the kink background is discussed.