Grassmann Coordinates Research Articles

A superspace formulation of the BV action

We show that the BV (Batalin-Vilkovisky) action, formulated with a generalized BRST symmetry (including the shift symmetry), is also invariant under a generalized anti-BRST transformation (where the antifields are the parameters of the transformation), when the gauge fixing lagrangian is both BRST and anti-BRST invariant. We show that for a general gauge fixing lagrangian, the BV action can be written in a manifestly generalized BRST invariant manner in a superspace with one Grassmann coordinate whereas it can be expressed in a manifestly generalized BRST and anti-BRST invariant manner in a superspace with two Grassmann coordinates when the gauge fixing lagrangian is invariant under both BRST and anti-BRST transformations.

UNIFICATION OF SPINS AND CHARGES IN GRASSMANN SPACE?

In a space of d(d > 5) ordinary and d Grassmann coordinates, fields manifest in an ordinary four-dimensional subspace as spinor (1/2, 3/2), scalar, vector or tensor fields with the corresponding charges, according to two kinds of generators of the Lorentz transformations in the Grassmann space. Vielbeins and spin connections define gauge fields-gravitational and Yang–Mills. For d = 15 the theory offers the unification of all known charges — spins and Yang–Mills charges — of fermionic and bosonic fields and all known interactions.

SCHIZOSYMMETRY: A NEW PARADIGM FOR SUPERFIELD EXPANSIONS

A new principle of ‘schizosymmetry’ is proposed for interpreting superfield expansions over Grassmann parameters in quantum field theory. Symmetry generators T phys , determining component field transformation properties, are allowed to depend on the grading via T phys =Tℙ e +T′ℙ o , where ℙ e and ℙ o are the projections onto even and odd components in the Grassmann coordinates, respectively, and T and T′ correspond to different representations. Examples are given for both internal and space-time schizosymmetry. In the latter case, superfield expansions have components with both integer and half-integer spin. A program for the Lagrangian realization of schizosymmetry, including unified model building, is outlined.

Grassmann coordinates and Lie algebras for unified models

A variety of Lie algebras and certain classes of representations can be constructed using Grassmann variables regarded as Lorentz scalar coordinates belonging to an internal space. The generators are realized as combinations of multilinear products of the coordinates and derivative operators, while the representations emerge as antisymmetric polynomials in the variables and are thus severely restricted. The nature of these realizations and the interconnections between various subalgebras, for N independent complex anticommuting coordinates, is explored. The addition of such Grassmann coordinates to the usual spacetime manifold provides a natural superfield setting for a unified theory of symmetries of elementary particles. The particle content can be further restricted by imposing discrete symmetries (Lie algebra automorphisms). For the case N=5 some anomaly free choices of multiplets are derived through the imposition of specific superfield duality conditions.

Twistor formulation of the heterotic D=10 superstring with manifest (8,0) world-sheet supersymmetry.

We propose a new formulation of the heterotic $D=10$ Green-Schwarz superstring whose worldsheet is a superspace with two even and eight odd coordinates. The action is manifestly invariant under both target-space supersymmetry and a worldsheet reparametrisation supergroup. It contains only a finite set of auxiliary fields. The key ingredient are the commuting spinor (twistor) variables, which naturally arise as worldsheet superpartners of the target space Grassmann coordinates. These spinors parametrise the sphere $S^8$ regarded as a coset space of the $D=10$ Lorentz group. The sphere is associated with the lightlike vector of one of the string Virasoro constraints. The origin of the on-shell $D=10$ fermionic kappa symmetry of the standard Green-Schwarz formulation is explained. An essential and unusual feature is the appearance of the string tension only on shell as an integration constant.

Differential calculus on the quantum superspace and deformation of phase space

We investigate non-commutative differential calculus on the supersymmetric version of quantum space where the non-commuting super-coordinates consist of bosonic as well as fermionic (Grassmann) coordinates. Multi-parametric quantum deformation of the general linear supergroup, $GL_q(m|n)$, is studied and the explicit form for the ${\hat R}$-matrix, which is the solution of the Yang-Baxter equation, is presented. We derive the quantum-matrix commutation relation of $GL_q(m|n)$ and the quantum superdeterminant. We apply these results for the $GL_q(m|n)$ to the deformed phase-space of supercoordinates and their momenta, from which we construct the ${\hat R}$-matrix of q-deformed orthosymplectic group $OSp_q(2n|2m)$ and calculate its ${\hat R}$-matrix. Some detailed argument for quantum super-Clifford algebras and the explict expression of the ${\hat R}$-matrix will be presented for the case of $OSp_q(2|2)$.

Lorentz-harmonic (super)fields and (super)particles

We apply the recently developed concept of a compact Lorentz-harmonic space, which is isomorphic to the sphere S D−2 , to the problem of covariant quantization of the superparticle in D = 3, 4, 6, 10. We study the structure of the representation of the Lorentz group realized on harmonic functions on S D−2 . The crucial difference between compact and non-compact harmonic analysis is explained. The massless harmonic fields depend on one space-time coordinate x ++ only, and on D−2 harmonic coordinates. It is shown how ordinary massless fields can be obtained from the harmonic ones by means of covariant integration on S D−2 . We construct a Lorentz-harmonic superspace, which involves only one quarter of the usual number of Grassmann coordinates. It closely resembles the light-cone superspace, however it is Lorentz covariant. This framework is used to formulate a superparticle action, in which all the constraints are first class, Lorentz-covariant and allow a straightforward canonical quantization.

Unified models from gauged supergroups. II. Exceptional series D(2,1;α), G(3), F(4)

Previously elaborated principles [J. Math. Phys. 31, 1783 (1990)] for dimensional reduction of models of gauged supergroups over supercoset spaces are applied to the exceptional superalgebras. The superspace is parametrized by a pair of Grassmann coordinates.

Unified models from gauged supergroups. I. Dimensional reduction formalism. Unitary and orthosymplectic series

Principles are elaborated for constructing consistent grand unified models (GUT’s) plus Higgs sectors from dimensional reduction of gauged supergroups over graded coset spaces. A search of possible GUT groups is made for unitary and orthosymplectic supergroups over a space parametrized by a single pair of Grassmann coordinates.

GRASSMANN WAVE FUNCTIONS AND INTRINSIC SPIN

By associating spin angular momentum with Sp(2) transformations on two Grassmann coordinates, we show how one may formulate spinor wave functions in complete analogy to spherical harmonics for orbital momentum. The relativistic generalization requires a doubling of Grassmann coordinates and a connection may be established with the Dirac equation.

String field theory with chiral coordinates

Interesting string theories, such as the heterotic string, involve chiral coordinates, that is coordinates which are either right-moving or left-moving on the world sheet. In this paper the field theoretic formulation of such theories is studied with most emphasis on the light-cone gauge method although the main techniques would also apply in the various covariant approaches to string field theory. In the heterotic string the internal symmetry is described in terms of eight left-moving complex bosonic coordinates in a manner which parallels the treatment of fermionic Grassmann coordinates. The wave functions, and hence the free string fields, are described in these coordinates and the three-string interaction is derived from the requirements of supersymmetry and Lorentz invariance both in its functional form and in terms of its oscillator modes.

Towards a manifestly supersymmetric formulation of N = 8 supergravity on a light-cone superspace

N = 8 supergravity is formulated in the light-cone gauge. An unconstrained superfield defined on a light-cone superspace is constructed from the component formulation. The light-cone superspace is spanned by Grassmann coordinates which belong to the spinor representation of SO(7), a subgroup of the transverse SO(9).

Null fermio-dynamics in superspace: Abelian and non-Abelian 3D massive vector fields

Half-order actions, i.e. actions where each term contains, at most, one spinorial derivative, of all the different known 3D supersymmetric massive vector systems are given. Since the two real Grassmann coordinates ( theta 1, theta 2)= theta of the 3D superspace respectively constitute the two independent (fermionic) light-front projections of theta along the null bosonic directions x+or-=21/2 (x0-or+x1), one of them, theta 1 is regarded as the spinorial time while the other one, theta 2, is seen as a (spinorial) spacelike variable. In terms of the corresponding null spinorial timelike and spinorial spacelike derivatives the field equations contain algebraic as well as differential constraints. The authors solve them for the different known 3D vector systems and obtain their corresponding unconstrained actions in superspace. In each case their evolution is shown to be controlled by a quantity which, by analogy with the standard bosonic case, is called the superenergy of the system. The analysis of the scalar case illustrates the connection between null dynamics in superspace and the standard null dynamics in 3D bosonic spacetime. It also shows how much the superenergy contributes to the null bosonic generator of the 3D dynamics.

Manifestly supersymmetric path integral formulation of the superstring field theories

Introducing a set of new superfields whose components are the space-time coordinates (momenta) and the Grassmann coordinates (momenta), we present a manifestly q −-supersymmetric path integral formulation of the free closed superstring field theory.

Classical spin and its quantization.

We develop a new formulation of the quantum mechanics of a relativistic fermion based on the existence of an extended set of underlying classical coordinates. We enlarge the classical theory to a superspace built out of ordinary space-time coordinates and a set of Grassmann coordinates which transform like Majorana spinors and describe a classical spin. We introduce a natural supersymmetry between the Grassmann and the space-time coordinates. We quantize the theory both canonically and via a relativistic particle-mechanics path-integral prescription. The quantization of the motion of the coordinates in the superspace gives rise to the standard Dirac structure of a relativistic quantum-mechanical particle with spin one-half.

Covariantly second-quantized string II

BRST invariance is used to second-quantize the interacting relativistic string. The zero-mode of the anticommuting string variables is identified as the Grassmann coordinate of BRST superfields. The massless sector is Yang-Mills theory in the usual Faddeev-Popov formalism.

Covariantly second-quantized string II

BRST invariance is used to second-quantize the interacting relativistic string. The zero-mode of the anticommuting string variables is identified as the Grassmann coordinate of BRST superfields. The massless sector is Yang-Mills theory in the usual Faddeev-Popov formalism.

Quantization of classical Grassman spin

We quantize the classical mechanics of anti-commuting Grassman coordinates which transform like Majorana spinors. We evaluate the Grassmann space Feynman path integral to obtain the standard Dirac propagator of a relativistic quantum-mechanical spin one-half particle. In order to obtain our result we impose a natural supersymmetry between ordinary space-time coordinates and the Grassmann coordinates.

A Note on Grassmann Coordinates (Joins and Intersections)

A Note on Grassmann Coordinates (Joins and Intersections)

The quadratic form positive definite on a linear manifold

1. Introduction. Necessary and sufficient conditions are established for a real quadratic form to be positive definite on a linear manifold, in a real vector space, explicit in terms of the dual Grassmann coordinates for the manifold.