Spacetime Algebra Research Articles

The Spacetime Algebra Approach to Massive Classical Electrodynamics with Magnetic Monopoles

Maxwell's equations with massive photons and magnetic monopoles are formulated using spacetime algebra. It is demonstrated that a single non-homogeneous multi-vectorial equation describes the theory. Two limiting cases are considered and their symmetries highlighted: massless photons with magnetic monopoles and finite photon mass in the absence of monopoles. Finally, it is shown that the EM-duality invariance is a symmetry of the Hamiltonian density (for Minkowskian spacetime) and Lagrangian density (for Euclidean 4-space) that reflects the signature of the respective metric manifold.

Relativistic particles and commutator algebras with twisted Poincaré transformation

Abstract In this paper we consider the massive and massless action for relativistic particle in D -dimensional flat space–time. We show that the Poincare space–time algebra in the commutator version, and the Killing field provides the generators of the Poincare algebra. We apply the non-commutative version to action, which is not Poincare invariant. This leads us to consider the twisted Poincare transformation, finally by using this transformation, we see that the action is invariant. By using the non-commutative space in massless action, in contrast to the commutative case the scale and conformal in-variance is broken by massive term [For an interesting discussion of the cosmological constant problem see A. Zee, Dark energy and the nature of the graviton, ].

Symmetric Spaces of Matter and Real Fermion Manifolds

The geometric algebra Cl3,1 generated by the Minkowski spacetime with signature {+++− } possesses a natural ternary partition which provides the Lie algebra of the standard model symmetry in an improved form. The symmetric spaces of matter embed a differentiable manifold of primitive idempotents which represents a real valued fermion space as an 8-dimensional real unitsphere in a 10-dimensional subspace with positive definite signature. The algebraic properties of the present theory of spacetime-matter are developed, beginning with the definiteness of the stabilizer algebra of neutrinos, investigating the orthogonality between fermions and neutrinos and ending with the curvature of the symmetric spaces of the strong force. The model brings together the quantum theory and relativity, as we conceive it at present, such that the standard model turns out to be a definite property of the spacetime algebra.

Doppler Shift from a Composition of Boosts with Thomas Rotation: a Spacetime Algebra Approach

As is well-known, Lorentz boosts do not form a group: the composite of two boosts is, in general, equivalent to the composite of a rotation — the so-called Thomas (or Wigner) rotation — and a single boost. Using spacetime algebra, the geometric algebra for Minkowski spacetime, we show in this paper how to relate the Thomas rotation to the Doppler shift experienced between two observers in relative motion. Both observers are moving away from a third observer, set in the laboratory, with relative non-parallel velocities.

New Lie-algebraic and quadratic deformations of Minkowski space from twisted Poincaré symmetries

We consider two new classes of twisted D=4 quantum Poincaré symmetries described as the dual pairs of noncocommutative Hopf algebras. Firstly we investigate a two-parameter class of twisted Poincaré algebras which provide the examples of Lie-algebraic noncommutativity of the translations. The corresponding associative star-products and new deformed Lie-algebraic Minkowski spaces are introduced. We discuss further the twist deformations of Poincaré symmetries generated by the twist with its carrier in Lorentz algebra. We describe corresponding deformed Poincaré group which provides the quadratic deformations of translation sector and define the quadratically deformed Minkowski space–time algebra.

Algebra of matter

Here it is shown that the known forces of nature unfold in parallel with an exact decomposition of the geometric algebra Cl 3,1 of spacetime. Up to an important common scalar this decomposition is a partition into a positive definite commutative graded space of strong forces and two negative non-commutative spaces with a quaternion structure for weak and other fields. The 6 fundamental spaces of strong forces are acted on by a rank 2 Lie algebra ℒ = sl Cl (2, ℝ) × so Cl (3, ℝ) of dimension 8 which brings in an isotropy group of the neutrinos, the flavour and colour symmetries and an isotropy of the weaker forces. The standard model, in an improved form, is a feature of the Clifford algebra of spacetime, and relativity as Lorentz invariance reduced to dimension (2, 1) is compatible with quantum theory. The whole Lorentz group cannot be, however, a property of physical motion. Due to the total exploitation of the whole geometric space and its convincing logical structure, the author believes there is at present no better algebraic model for the totality of known symmetries of physical dynamics.

Basic systems of orthogonal functions for space-time multivectors

Space-time multivectors in Clifford algebra (space-time algebra) and their application to nonlinear electrodynamics are considered. Functional product and infinitesimal operators for translation and rotation groups are introduced, where unit pseudoscalar or hyperimaginary unit is used instead of imaginary unit. Basic systems of orthogonal functions (plane waves, cylindrical, and spherical) for space-time multivectors are built by using the introduced infinitesimal operators. Appropriate orthogonal decompositions for electromagnetic field are presented. These decompositions are applied to nonlinear electrodynamics. Appropriate first order equation systems for cylindrical and spherical radial functions are obtained. Plane waves, cylindrical, and spherical solutions to the linear electrodynamics are represented by using the introduced orthogonal functions. A decomposition of a plane wave in terms of the introduced spherical harmonics is obtained.

Relativity in Clifford's Geometric Algebras of Space and Spacetime

Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships between formulations of special relativity in the spacetime algebra (STA) Cl{1,3} of Minkowski space, and in the algebra of physical space (APS)Cl{3}. STA lends itself to an absolute formulation of relativity, in which paths, fields, and other physical properties have observer-independent representations. Descriptions in APS are related by a one-to-one mapping of elements from APS to the even subalgebra STA+ of STA. With this mapping, reversion in APS corresponds to hermitian conjugation in STA. The elements of STA+ are all that is needed to calculate physically measurable quantities because only they entail the observer dependence inherent in any physical measurement. As a consequence, every relativistic physical process that can be modeled in STA also has a representation in APS, and vice versa. In the presence of two or more inertial observers, two versions of APS present themselves. In the absolute version, both the mapping to STA+ and hermitian conjugation are observer dependent, and the proper basis vectors are persistent vectors that sweep out timelike planes. In the relative version, the mapping and hermitian conjugation are then the same for all observers. Relative APS gives a covariant representation of relativistic physics with spacetime multivectors represented by multiparavectors. We relate the two versions of APS as consistent models within the same algebra.

A non-commutative Minkowskian spacetime from a quantum AdS algebra

A quantum deformation of the conformal algebra of the Minkowskian spacetime in (3+1) dimensions is identified with a deformation of the (4+1)-dimensional AdS algebra. Both Minkowskian and AdS first-order non-commutative spaces are explicitly obtained, and the former coincides with the well-known κ-Minkowski space. Next, by working in the conformal basis, a new non-commutative Minkowskian spacetime is constructed through the full (all orders) dual quantum group spanned by deformed Poincaré and dilation symmetries. Although Lorentz invariance is lost, the resulting non-commutative spacetime is quantum group covariant, preserves space isotropy and, furthermore, can be interpreted as a generalization of the κ-Minkowski space in which a variable fundamental scale (Planck length) appears.

Dynamic and Geometric Phase Formulas in the Hestenes-Dirac Theory

We examine the dynamic and geometric phases of the electron in quantum mechanics using Hestenes' spacetime algebra formalism. First the standard dynamic phase formula is translated into the spacetime algebra. We then define new formulas for the dynamic and geometric phases that can be used in Hestenes' formalism.

Traditional vectors as an introduction to geometric algebra

The 2002 Oersted Medal Lecture by David Hestenes concerns themany advantages for education in physics if geometric algebra were toreplace standard vector algebra. However, such a change has difficultiesfor those who have been taught traditionally. A new way of introducinggeometric algebra is presented here using a four-element array composed oftraditional vector and scalar products. This leads to an explicit 4 × 4matrix representation which contains key requirements for three-dimensionalgeometric algebra. The work can be extended to include Maxwell’s equationswhere it is found that curl and divergence appear naturally together.However, to obtain an explicit representation of space–time algebrawith the correct behaviour under Lorentz transformations, an 8 × 8matrix representation has to be formed. This leads to a Dirac representation ofMaxwell’s equations showing that space–time algebra has hidden within itsformalism the symmetry of ‘parity, charge conjugation and time reversal’.

Spacetime physics with geometric algebra

This is an introduction to spacetime algebra (STA) as a unified mathematical language for physics. STA simplifies, extends, and integrates the mathematical methods of classical, relativistic, and quantum physics while elucidating geometric structure of the theory. For example, STA provides a single, matrix-free spinor method for rotational dynamics with applications from classical rigid body mechanics to relativistic quantum theory—thus significantly reducing the mathematical and conceptual barriers between classical and quantum mechanics. The entire physics curriculum can be unified and simplified by adopting STA as the standard mathematical language. This would enable early infusion of spacetime physics and give it the prominent place it deserves in the curriculum.

More features of the mythic spacetime algebra

This is the third of a trio of papers on mythic spacetime (see Griffin 2000, 2001). The elementary operations of mathematics are used to link mythology, literature, language, and psychology. The operation of multiplication is redefined as association. Equations utilize variables from sets, with members such as who, what, where, and when. The basic framework consists of a field algebra nested in paired functions. The functions index sets with either of two states. A hierarchical order of operations provides a range to situate different concepts of theme, symbol, and motif.

Clifford Algebra of Spacetime and the Conformal Group

We demonstrate the emergence of the conformal group SO(4,2) from the Clifford algebra of spacetime. The latter algebra is a manifold, called Clifford space, which is assumed to be the arena in which physics takes place. A Clifford space does not contain only points (events), but also lines, surfaces, volumes, etc., and thus provides a framework for description of extended objects. A subspace of the Clifford space is the space whose metric is invariant with respect to the conformal group SO(4,2) which can be given either passive or active interpretation. As advocated long ago by one of us, active conformal transformations, including dilatations, imply that sizes of physical objects can change as a result of free motion, without the presence of forces. This theory is conceptually and technically very different from Weyl's theory and provides when extended to a curved conformal space a resolution of the long standing problem of realistic masses in Kaluza—Klein theories.

Lorentz-Invariant Non-Commutative Space-Time Based on DFR Algebra

It is argued that the familiar algebra of the non-commutative space-time with $c$-number $\theta^{\mu\nu}$ is inconsistent from a theoretical point of view. Consistent algebras are obtained by promoting $\theta^{\mu\nu}$ to an anti-symmetric tensor operator ${\hat\theta}^{\mu\nu}$. The simplest among them is Doplicher-Fredenhagen-Roberts (DFR) algebra in which the triple commutator among the coordinate operators is assumed to vanish. This allows us to define the Lorentz-covariant operator fields on the DFR algebra as operators diagonal in the 6-dimensional $\theta$-space of the hermitian operators, ${\hat\theta}^{\mu\nu}$. It is shown that we then recover Carlson-Carone-Zobin (CCZ) formulation of the Lorentz-invariant non-commutative gauge theory with no need of compactification of the extra 6 dimensions. It is also pointed out that a general argument concerning the normalizability of the weight function in the Lorentz metric leads to a division of the $\theta$-space into two disjoint spaces not connected by any Lorentz transformation so that the CCZ covariant moment formula holds true in each space, separately. A non-commutative generalization of Connes' two-sheeted Minkowski space-time is also proposed. Two simple models of quantum field theory are reformulated on $M_4\times Z_2$ obtained in the commutative limit.

Superstring Vacua of 4-dimensional PP-Waves with Enhanced Supersymmetry

We study the superstring vacua constructed from the conformal field theories of the type H_4 x M, where H_4 denotes the super Nappi-Witten model (super WZW model on the 4-dimensional Heisenberg group H_4) and M denotes an arbitrary N=2 rational superconformal field theory with c=9. We define (type II) superstring vacua with 8 supercharges, which are twice as many as those on the backgrounds of H_4 x CY_3. We explicitly construct as physical vertices the space-time SUSY algebra that is a natural extension of H_4 Lie algebra. The spectrum of physical states is classified into two sectors: (1) strings freely propagating along the transverse plane of pp-wave geometry and possessing the integral U(1)_R-charges in M sector, and (2) strings that do not freely propagate along the transverse plane and possess the fractional U(1)_R-charges in M. The former behaves like the string excitations in the usual Calabi-Yau compactification, but the latter defines new sectors without changing the physics in ``bulk'' space. We also analyze the thermal partition functions of these systems, emphasizing the similarity to the DLCQ string theory. As a byproduct we prove the supersymmetric cancellation of conformal blocks in an arbitrary unitary N=2 SCFT of c=12 with the suitable GSO projection.

FREE FIELD EQUATIONS FOR SPACE–TIME ALGEBRAS WITH TENSORIAL MOMENTUM

Free field equations, with various spins, for space–time algebras with second-rank tensor (instead of the usual vector) momentum are constructed. Similar algebras are appearing in superstring/M theories. Special attention is paid to gauge invariance properties, in particular the spin-two equations with gauge invariance are constructed for dimensions 2+2 and 2+4, and the connection with Einstein equation and diffeomorphism invariance is established.

Super-pp-wave algebra from super-AdS× S algebras in eleven dimensions

Maximally supersymmetric spacetime algebras in eleven dimensions, which are the isometry superalgebras of Minkowski space, AdS 7× S 4, AdS 4× S 7 and pp-wave background, are related by Inönü–Wigner contractions. The super-AdS 4(7)× S 7(4) algebras allow to introduce two contraction parameters, the one for the flat limit to the super-Poincaré algebra and the other for a Penrose limit to the super-pp-wave algebra. Under these contractions supersymmetries are maintained because the Jacobi identity of three supercharges holds for any values of contraction parameters.

Superstrings on PP-Wave Backgrounds and Symmetric Orbifolds

We study the superstring theory on pp-wave background with NSNS-flux that is realized as the Penrose limit of AdS_3 x S^3 x M^4, where M^4 is T^4 or T^4/Z_2(~ K3). Quantizing this system in the covariant gauge, we explicitly construct the space-time supersymmetry algebra and the complete set of DDF operators. We analyse the spectrum of physical states by using the spectrally flowed representations of current algebra. This spectrum is classified by the ``short string sectors'' and the ``long string sectors'' as in AdS_3 string theory. The states of the latter propagate freely along the transverse plane of pp-wave background, but the states of the former do not. We compare the short string spectrum with the BPS and almost BPS states which have large R-charges in the symmetric orbifold conformal theory, which is known as the candidate of dual theory of superstrings on AdS_3 x S^3 x M^4. We show that every short string states can be embedded successfully in the single particle Hilbert space of symmetric orbifold conformal theory.

Comment on `Dirac theory in spacetime algebra'

In contrast to formulations of the Dirac theory by Hestenes and by the current author, the formulation recently presented by W. P. Joyce [J. Phys. A: Math. Gen. 34 (2001) 1991--2005] is equivalent to the usual Dirac equation only in the case of vanishing mass. For nonzero mass, solutions to Joyce's equation can be solutions either of the Dirac equation in the Hestenes form or of the same equation with the sign of the mass reversed, and in general they are mixtures of the two possibilities. Because of this relationship, Joyce obtains twice as many linearly independent plane-wave solutions for a given momentum eigenvalue as exist in the conventional theory. A misconception about the symmetry of the Hestenes equation and the geometric significance of the algebraic spinors is also briefly discussed.