Spacetime Algebra Research Articles

Reformulation of elliptic equations for heat transfer and diffusion in solids with space-time algebra

In this paper, we demonstrate the application of non-commutative space-time algebra of sedeons to generalize the system of equations describing heat transfer and impurity diffusion in solids at finite velocity. It is shown that by analogy with electrodynamics, these transfer processes can be described using a compact second-order sedeonic equation for generalized scalar and vector potentials. On the one hand, this equation is reduced to the system of first-order differential equations for vortex-less mass and heat flows, and on the other hand, it can be transformed to the second-order elliptical equations for the profiles of temperature and impurity concentration. The comparison of peculiarities in transfer within the frames of parabolic and elliptic equations is discussed.

Explicit Baker–Campbell–Hausdorff–Dynkin formula for spacetime via geometric algebra

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations [Formula: see text] in the spin representation (a.k.a. Lorentz rotors) in terms of their generators [Formula: see text]: [Formula: see text] This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension [Formula: see text], naturally generalizing Rodrigues’ formula for rotations in [Formula: see text]. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex [Formula: see text] matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic [Formula: see text]-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

A force proportional to velocity squared derived from spacetime algebra

ABSTRACT The underlying geometry of spacetime algebra allows one to derive a force by contracting the relativistic generalization of angular momentum, M, with the mass-current, mω, where w is a proper 4-vector velocity. By applying this force to a cosmological object, a repulsive inverse distance-square law is found, which is proportional to the velocity dispersion squared of that structure. It is speculated if this finding may be relevant to the recent suggestion, that such a force may accelerate the expanding universe with no need for a cosmological constant.

Uncertainty Principle for Space–Time Algebra-Valued Functions

In this paper, we present a set of important properties of the special relativistic Fourier transformation (SFT) on the complex space–time algebra $${\mathcal {G}}{(3,1)}$$ , such as inversion property, the Plancherel theorem, and the Hausdorff–Young inequality. The main objective of this article is to introduce the concept of the vector derivative in geometric algebra and using it together with the notion of the space–time split to derive the Heisenberg–Pauli–Weyl inequality. Finally, we apply the SFT properties for proving the Donoho–Stark uncertainty principle for $${\mathcal {G}}{(3,1)}$$ multi-vector functions.

Complexifying the Spacetime Algebra by Means of an Extra Timelike Dimension: Pin, Spin and Algebraic Spinors

Because of the isomorphism \(C \ell _{1,3}({\mathbb {C}})\cong C \ell _{2,3}({\mathbb {R}})\), it is possible to complexify the spacetime Clifford algebra \(C \ell _{1,3}({\mathbb {R}})\) by adding one additional timelike dimension to the Minkowski spacetime. In a recent work we showed how this treatment provide a particular interpretation of Dirac particles and antiparticles in terms of the new temporal dimension. In this article we thoroughly study the structure of the real Clifford algebra \(C \ell _{2,3}({\mathbb {R}})\) paying special attention to the isomorphism \(C \ell _{1,3}({\mathbb {C}})\cong C \ell _{2,3}({\mathbb {R}})\) and the embedding \(C \ell _{1,3}({\mathbb {R}})\subseteq C \ell _{2,3}({\mathbb {R}})\). On the first half of this article we analyze the Pin and Spin groups and construct an injective mapping \({\text {Pin}}(1,3)\hookrightarrow {\text {Spin}}(2,3)\), obtaining in particular elements in \({\text {Spin}}(2,3)\) that represent parity and time reversal. On the second half of this paper we study the spinor space of the algebra and prove that the usual structure of complex spinors in \(C \ell _{1,3}({\mathbb {C}})\) is reproduced by the Clifford conjugation inner product for real spinors in \(C \ell _{2,3}({\mathbb {R}})\).

ℱ(P) quantum mechanics

We explore the possibility to extend the Heisenberg’s uncertainty principle to a nonlinear extension of the quantum algebra related to a functional operator of the momenta as [Formula: see text]. We show that such an extension of quantum mechanics is intimately connected to the non-commutative space-time algebra and the Lorentz symmetry deformations. We show that a large class of [Formula: see text] models can introduce superluminal modes in the quantized theories. We also show that the Hořava–Lifshitz theory is related to a large class of [Formula: see text] Quantum Mechanics.

Parabolic Verma Modules and Invariant Differential Operators

In the present paper we continue the project of systematic classification and construction of invariant differential operators for non-compact semisimple Lie groups. This time we make the stress on one of the main building blocks, namely the Verma modules and the corresponding parabolic subalgebras. In particular, we start the study of the relation between the parabolic subalgebras of real semisimple Lie algebras and of their complexification. Two cases are given in more detail: the conformal algebra of 4D Minkowski space-time and the minimal parabolics of classical real semisimple Lie algebras.

Sedeonic Equations in Field Theory

We consider the factorization of differential operator of Klein–Gordon equation on the base of space-time algebra of sixteen-component sedeons. It is shown that generalized sedeonic wave equations can be used both to describe quantum particles and the force fields responsible for the interaction of particles. In particular, we discuss the first-order and second-order wave equations for sedeonic potentials describing fields with zero and nonzero mass of quantum. We demonstrate the application of sedeonic space-time operators to describe quantum particles and in particular the calculation of energy spectrum of electron in an external magnetic field. The relations between the sedeons and the spinors are discussed.

Two θμν -deformed covariant relativistic quantum phase spaces as Poincaré-Hopf algebroids

We consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known $\theta_{\mu \nu }$-deformed $D=4$ Poincare-Hopf algebra $\mathbb{H}$. The first algebroid describes $\theta_{\mu \nu }$-deformed relativistic phase space with canonical NC space-time (constant $\theta_{\mu \nu }$ parameters) and the second one incorporates dual to $\mathbb{H}$ quantum $\theta_{\mu \nu }$-deformed Poincare-Hopf group algebra $\mathbb{G}$, which contains noncommutative space-time translations given by $\Lambda $-dependent $\Theta_{\mu \nu }$ parameters ($% \Lambda $ $\equiv \Lambda_{\mu \nu }$ parametrize classical Lorentz group). The canonical $\theta_{\mu \nu }$-deformed space-time algebra and its quantum phase space extension is covariant under the quantum Poincare transformations described by $\mathbb{G}$. We will also comment on the use of Hopf algebroids for the description of multiparticle structures in quantum phase spaces.

On Bishop frame of a pseudo null curve in Minkowski space-time

In this paper, we introduce the Bishop frame of a pseudo null curve $\alpha$ in Minkowski space-time. We obtain the Bishop frame's equations and the relation between the Frenet frame and the Bishop frame. We find the third order nonlinear differential equation whose particular solutions determine the form of the Bishop curvatures. By using space-time geometric algebra, we derive the Darboux bivectors $D$ and $\tilde{D}$ of the Frenet and the Bishop frame of $\alpha$, respectively. We give geometric interpretations of the Frenet and the Bishop curvatures of $\alpha$ in terms of areas of the projections of the corresponding Darboux bivectors onto the planes spanned by the frame vector's fields.

Geographic wayfinders and space-time algebra

Time Geography is a framework for describing reachable points in a (static) spatio-temporal environment. While originally devised to facilitate reasoning about an individual's or a population's living conditions, it was later adapted to many other applications. A wayfinder is an entity that moves through a space-time continuum with possible obstacles. We show how to model the pertinent notions in relational algebra (and, more abstractly, in modal semirings) with box and diamond operators. Admissible or undesired regions can be described as Boolean combinations of primitive regions such as the set of all points reachable by forward or backward movement from a given region or starting point. To derive results about the region blocked by the union of two regions we introduce an abstract algebraic view of coordinates that is largely independent of dimensional and metric aspects and thus very general. Moreover, the approach lends itself quite well to machine-supported proofs.

Clifford Algebras, Multipartite Systems and Gauge Theory Gravity

In this paper we present a multipartite formulation of gauge theory gravity based on the formalism of space-time algebra for gravitation developed by Lasenby and Doran (Lasenby, A. N., Doran, C. J. L, and Gull, S.F.: Gravity, gauge theories and geometric algebra. Phil. Trans. R. Soc. Lond. A, 582, 356:487 (1998)). We associate the gauge fields with description of fermionic and bosonic states using the generalized graded tensor product. Einstein's equations are deduced from the graded projections and an algebraic Hopf-like structure naturally emerges from formalism. A connection with the theory of the quantum information is performed through the minimal left ideals and entangled qubits are derived. In addition applications to black holes physics and standard model are outlined.

Reformulation of Multifluid Plasma Equations in Terms of Spacetime Algebra

Using the exact isomorphism between multifluid plasma theory and classical electromagnetism, we apply the theorems and techniques of spacetime algebra to the multifluid equations of a plasma in order to present a new formulation in a compact and elegant way. A set of Maxwell equations for a plasma is combined into a single equation similar to their counterparts in electromagnetism, linear gravitation and fluid mechanics. The presented formulation also proposes a useful method for the generalization of the plasma wave equation.

Sedeonic Field Equations for Dyons

We discuss the theoretical description of dyons having simultaneously both electric and magnetic charges on the basis of space-time algebra of 16-component sedeons. We show that the generalized sedeonic equations for electromagnetic field of dyons can be reformulated in equivalent canonical form as the equations for redefined field potentials, field strengths and sources. The relations for energy and momentum as well as the relations for Lorentz invariants of dyonic electromagnetic field are derived. Additionally, we discuss the sedeonic second-order and first-order wave equations describing the quantum behavior of dyons in an external dyonic electromagnetic field.

Continuous-spin mixed-symmetry fields in AdS(5)

Free mixed-symmetry continuous-spin fields propagating in AdS(5) space and flat R(4,1) space are studied. In the framework of a light-cone gauge formulation of relativistic dynamics, we build simple actions for such fields. The realization of relativistic symmetries on the space of light-cone gauge mixed-symmetry continuous-spin fields is also found. Interrelations between constant parameters entering the light-cone gauge actions and eigenvalues of the Casimir operators of space-time symmetry algebras are obtained. Using these interrelations and requiring that the field dynamics in AdS(5) be irreducible and classically unitary, we derive restrictions on the constant parameters and eigenvalues of the second-order Casimir operator of the algebra.

AdS2 \xd7 S6 versus AdS6 \xd7 S2 in Type IIB supergravity

We obtain the complete local solutions with 16 supersymmetries to Type IIB supergravity on a space-time of the form AdS2 × S6 warped over a Riemann surface Σ in terms of two locally holmorphic functions on Σ. We construct the general Ansatz for the bosonic supergravity fields and supersymmetry generators compatible with the SO(2, 1) ⊕ SO(7) isometry algebra of space-time, which extends to the corresponding real form of the exceptional Lie superalgebra F (4). We reduce the BPS equations to this Ansatz, obtain their general local solutions, and show that these local solutions solve the full Type IIB supergravity field equations and Bianchi identities. We contrast the AdS2 × S6 solution with the closely related AdS6 × S2 case and present our results for both in parallel. Finally, we present a preliminary analysis of positivity and regularity conditions for AdS2 × S6, but postpone the construction of globally regular solutions to a subsequent paper.

Discretization of Maxwell’s Equations for Non-inertial Observers Using Space-Time Algebra

We employ classical Maxwell’s equations formulated in space-time algebra to perform discretization of moving geometries directly in space-time. All the derivations are carried out without any non-relativistic assumptions, thus the application area of the scheme is not restricted to low velocities. The 4D mesh construction is based on a 3D mesh stemming from a conventional 3D mesh generator. The movement of the system is encoded in the 4D mesh geometry, enabling an easy extension of well-known 3D approaches to the space-time setting. As a research example, we study a manifestation of Sagnac’s effect in a rotating ring resonator. In case of constant rotation, the space-time approach enhances the efficiency of the scheme, as the material matrices are constant for every time step, without abandoning the relativistic framework.

Sedeonic equations of ideal fluid

In the present paper, we propose the generalized equations for an ideal fluid based on space-time algebra of sixteen-component sedeons. It is shown that the dynamics of isentropic fluid can be described by sedeonic first-order wave equation for fluid potentials. The key features of the proposed formalism are illustrated on the problem of the sound waves propagation. We consider the plane wave solution of linearized sedeonic wave equation and derive the second-order relations for the sound potential analogues to the Poynting theorem in electrodynamics. The generalization of proposed sedeonic equations for the description of viscous fluid is also discussed.

Spacetime algebra for the reformulation of fluid field equations

In the light of the analogy between electromagnetism and fluid dynamics, the Maxwell-type equations of compressible fluids are reformulated on the basis of spacetime algebra. In this paper, it is proved that this algebra provides an efficient mathematical tool for describing fluid fields in a compact and elegant way. Moreover, the fluid wave equation in terms of potentials are derived in a form similar to electromagnetic and gravitational counterparts previously derived using spacetime algebra.

Generalized Klein–Gordon field equations with octonion space-time (OST) algebra

The concept of space-time representation is redefined using the octonion space-time (OST) algebra. In this study, describing the properties of octonions and their possible connection with Euclidean space-times, the internal and external space-time events are represented within the OST algebra. Keeping in mind the octonionic dual-Euclidean space-times, we express the homogeneous field equations which leads to the symmetrical nature of internal and external space-times. We derive the generalized Proca–Maxwell equations for massive-dyons in the case of the OST algebra. Accordingly, we have obtained a new set of octonionic Klein–Gordon potential (KGP) and Klein–Gordon field (KGF) equations for massive dyons from the generalized Proca–Maxwell equations. This formalism demonstrates that the octonionic KGP and KGF equations can be expressed in a single equation and it is equivalent to energy-momentum relation for dyons. As such, we have made an attempt to write the conservation of Noetherian current from the octonionic Klein–Gordon equations.