Spacetime Algebra Research Articles

A twist at infinite distance in the CHL string

We analyze a space-time algebra of BPS states that emerges in the infinite distance limit in the moduli space of the nine-dimensional CHL string as the theory decompactifies to the ten-dimensional E8 × E8 heterotic string. We find an affine algebra as expected from the heterotic case, but in a twisted version.

Quantum number conservation: a tool in the design and analysis of high energy experiments

The Clifford Unification algebra Cl 7,7 is shown to provide a unified description of all the elementary fermions and hadrons in terms of seven binary quantum numbers. Four of these are defined in existing theories, namely spin-direction, fermion/anti-fermion pairing and the two commuting generators in the SU(3) description of quark colour. A Cl 3,3 sub-algebra integrates the Cl 1,3 space-time and Dirac algebras providing a new definition of time direction and identifying parity as a new quantum number. A further two quantum numbers are related to the commuting generators of an SU(3) description of fermion generations. All seven are conserved in the decays and interactions of charged particles, suggesting that quantum number conservation provides a useful tool in the design and interpretation of high energy experiments. This is especially relevant when neutral particles and parity are involved.

On the Continuity Equation in Space–Time Algebra: Multivector Waves, Energy–Momentum Vectors, Diffusion, and a Derivation of Maxwell Equations

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.’s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system’s structure invariant, a system with the structure of Maxwell’s field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations.

Beyond Wilson? Carroll from current deformations

At extreme energies, both low and high, the spacetime symmetries of relativistic quantum field theories (QFTs) are expected to change with Galilean symmetries emerging in the very low energy domain and, as we will argue, Carrollian symmetries appearing at very high energies. The formulation of Wilsonian renormalisation group seems inadequate for handling these changes of the underlying Poincare symmetry of QFTs and it seems unlikely that these drastic changes can be seen within the realms of relativistic QFT. We show that contrary to this expectation, changes in the spacetime algebra occurs at the very edges of parameter space. In particular, we focus on the very high energy sector and show how bilinears of U(1) currents added to a two dimensional (massless) scalar field theory deform the relativistic spacetime conformal algebra to conformal Carroll as the effective coupling of the deformation is dialed to infinity. We demonstrate this using both a symmetric and an antisymmetric current-current deformation for theories with multiple scalar fields. These two operators generate distinct kinds of quantum flows in the coupling space, the symmetric driven by Bogoliubov transformations and the antisymmetric by spectral flows, both leading to Carrollian CFTs at the end of the flow.

From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited

Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our application of geometric (Clifford) algebras (GAs) in quantum computing as originally presented in [C. Cafaro and S. Mancini, Adv. Appl. Clifford Algebras 21, 493 (2011)]. Our focus is on testing the usefulness of geometric algebras (GAs) techniques in two quantum computing applications. First, making use of the geometric algebra of a relativistic configuration space (namely multiparticle spacetime algebra or MSTA), we offer an explicit algebraic characterization of one- and two-qubit quantum states together with a MSTA description of one- and two-qubit quantum computational gates. In this first application, we devote special attention to the concept of entanglement, focusing on entangled quantum states and two-qubit entangling quantum gates. Second, exploiting the previously mentioned MSTA characterization together with the GA depiction of the Lie algebras SO3;R and SU2;C depending on the rotor group Spin+3,0 formalism, we focus our attention to the concept of universality in quantum computing by reevaluating Boykin’s proof on the identification of a suitable set of universal quantum gates. At the end of our mathematical exploration, we arrive at two main conclusions. Firstly, the MSTA perspective leads to a powerful conceptual unification between quantum states and quantum operators. More specifically, the complex qubit space and the complex space of unitary operators acting on them merge in a single multivectorial real space. Secondly, the GA viewpoint on rotations based on the rotor group Spin+3,0 carries both conceptual and computational advantages compared to conventional vectorial and matricial methods.

Symbolic studies of Maxwell’s equations in space-time algebra formalism

Different implementations of Clifford algebra: spinors, quaternions, and geometric algebra, are used to describe physical and technical systems. The geometric algebra formalism is a relatively new approach, destined to be used primarily by engineers and applied researchers. In a number of works, the authors examined the implementation of the geometric algebra formalism for computer algebra systems. In this article, the authors extend elliptic geometric algebra to hyperbolic space-time algebra. The results are illustrated by different representations of Maxwell’s equations. Using a computer algebra system, Maxwell’s vacuum equations in the space-time algebra representation are converted to Maxwell’s equations in vector formalism. In addition to practical application, the authors would like to draw attention to the didactic significance of these studies.

Clifford algebra Cl(0,6) approach to beyond the standard model and naturalness problems

Is there more to Dirac’s gamma matrices than meets the eye? It turns out that gamma zero can be factorized into a product of three operators. This revelation facilitates the expansion of Dirac’s space-time algebra to Clifford algebra [Formula: see text]. The resultant rich geometric structure can be leveraged to establish a combined framework of the standard model and gravity, wherein a gravi-weak interaction between the vierbein field and the extended weak gauge field is allowed. Inspired by the composite Higgs model, we examine the vierbein field as an effective description of the fermion–antifermion condensation. The compositeness of space-time manifests itself at an energy scale which is different from the Planck scale. We propose that all the regular classical Lagrangian terms are of quantum condensation origin, thus possibly addressing the cosmological constant problem provided that we exercise extreme caution in the renormalization procedure that entails multiplications of divergent integrals. The Clifford algebra approach also permits a weaker form of charge conjugation without particle–antiparticle interchange, which leads to a Majorana-type mass that conserves lepton number. Additionally, in the context of spontaneous breaking of two global [Formula: see text] symmetries, we explore a three-Higgs-doublet model which could explain the fermion mass hierarchies.

The Beurling Theorem in Space–Time Algebras

The Beurling Theorem in Space–Time Algebras

Notes on the L∞-approach to local gauge field theories

It is well known that a Q-manifold gives rise to an L∞-algebra structure on the tangent space at a fixed point of the homological vector field. From the field theory perspective this implies that the expansion of a classical Batalin-Vilkovisky (BV) formulation around a vacuum solution can be equivalently cast into the form of an L∞-algebra. In so doing, the BV symplectic structure determines a compatible cyclic structure on the L∞-algebra. Moreover, L∞ quasi-isomorphisms correspond to so-called equivalent reductions (also known as the elimination of generalized auxiliary fields) of the respective BV systems. Although at the formal level the relation is straightforward, its implementation in field theory requires some care because the underlying spaces become infinite-dimensional. In the case of local gauge theories, the relevant spaces can be approximated by nearly finite-dimensional ones through employing the jet-bundle technique. In this work we study the L∞-counterpart of the jet-bundle BV approach and generalise it to a more flexible setup of so-called gauge PDEs. It turns out that in the latter case the underlying L∞-structure is analogous to that of Chern-Simons theory. In particular, the underlying linear space is a module over the space-time exterior algebra and the higher L∞-maps are multilinear. Furthermore, a counterpart of the cyclic structure turns out to be degenerate and possibly nonlinear, and corresponds to a compatible presymplectic structure the gauge PDE, which is known to encode the BV symplectic structure and hence the full-scale Lagrangian BV formulation. Moreover, given a degenerate cyclic structure one can consistently relax the L∞-axioms in such a way that the formalism still describes non-topological models but involves only finitely-generated modules, as we illustrate in the example of Yang-Mills theory.

ON SPACETIME ALGEBRA AND ITS RELATIONS WITH NEGATIVE MASSES

We consider four subsets of the complexified spacetime algebra, namely the real even part, the real odd part, the imaginary even part and the imaginary odd part. This naturally leads to the four connected components of the Lorentz group, supplemented each time by an additional symmetry. We then examine how these four parts impact the Dirac equation and show that four types of matter arise with positive and negative masses as well as positive and negative charges.

Gauging the Maxwell Extended GLn,R and SLn+1,R Algebras

We consider the extension of the general-linear and special-linear algebras by employing the Maxwell symmetry in D space-time dimensions. We show how various Maxwell extensions of the ordinary space-time algebras can be obtained by a suitable contraction of generalized algebras. The extended Lie algebras could be useful in the construction of generalized gravity theories and the objects that couple to them. We also consider the gravitational dynamics of these algebras in the framework of the gauge theories of gravity. By adopting the symmetry-breaking mechanism of the Stelle–West model, we present some modified gravity models that contain the generalized cosmological constant term in four dimensions.

Phenomenology from Dirac Equation with Euclidean-Minkowskian “Gravity Phase”

Over the past decades, many authors advertised models on complexified spacetime algebras for use in describing gravity. This work aims at providing phenomenological support to such claims, by introducing a one-parameter real phase $\alpha$ to the conventional Dirac equation with $\frac{1}{r}$-type potential. This phase allows to transition between Euclidean ($\alpha=0,\pm\pi,\pm2\pi,\ldots$) and Minkowskian ($\alpha=\pm\frac{\pi}{2},\pm\frac{3\pi}{2},\ldots$) geometry, as two distinct cases that one may expect from some complexified spacetime. The configuration space is modeled on $4\times4$ matrix algebra over the bicomplex numbers, $\mathbb{C}\oplus\mathbb{C}$. Spin-$\frac{1}{2}$ Coulomb scattering (Rutherford scattering) in Born approximation is then executed. All calculations are done ``from scratch'', as they could have been done some 85 years ago. By removing elegance from field theory that has since become customary, this paper aims at remaining as generally applicable as possible, for a wide range of candidate models that contain such a phase $\alpha$ in one way or another. Results for backscattering and cross section at high energies are compared with results from General Relativity calculations. Effects on intergalactic gas distribution and momentum transfer from scattering high-energy leptons are sketched.

Some recent results for SU(3) and octonions within the geometric algebra approach to the fundamental forces of nature

Different ways of representing the group within a Geometric Algebra approach are explored. As part of this, we consider characteristic multivectors for and how these are linked with decomposition of generators into commuting bivectors. The setting for this work is within a 6d Euclidean Clifford Algebra. We then go on to consider whether the fundamental forces of particle physics might arise from symmetry considerations in just the 4d geometric algebra of spacetime—the STA. As part of this, a representation of is found wholly within the STA, involving preservation of a bivector norm. We also show how Octonions can be fully represented within the Spacetime Algebra, which we believe will be useful in making them understandable and accessible to a new community in Physics and Engineering. The two strands of the paper are drawn together in showing how preserving the octonion norm is the same as preserving the timelike part of the Dirac current of a particle. This suggests a new model for the symmetries preserved in particle physics. Following on from work by Günaydin and Gürsey on the link between quarks, and octonions, and by Furey on chains of octonionic multiplications, we show how both of these fit well within our scheme and give some wholly STA versions of the operations involved, which in the cases considered have easily understandable equivalents in terms of 4d geometry. Links with larger groups containing , such as and , are also considered.

The Lorentz Group Using Conformal Geometric Algebra and Split Quaternions for Color Image Processing: Theory and Practice

The processing of color images is of great interest, because the human perception of color is a very complex process, still not well understood. In this article, firstly the authors present an analysis of the well-known mathematical methods used to model color as a phenomenon and to process color images. We propose an adequate metric to process color images using the Minkowski or space-time metric. We propose that the processing of HSV images can be done using the HSV cone in the quaternion split algebra or the conformal geometric algebra frameworks. We show that the processing of RGB images using the Euclidean metric in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {R}^{3}$ </tex-math></inline-formula> doesn’t yield good results. The colors of images change by daylight, we understand this phenomenon as the color change follows the action of the Lorentz Lie group. Consequently, we formulate a new model for representing and processing color using split quaternions and space-time conformal geometric algebra. We propose new algorithms for practical color image processing: we formulate the novel Split Quaternion Fourier Transform for color image processing and we interpolate color images using Split Motors which belong to the space-time conformal geometric algebra. The experimental part proves that real color images have to be processed in the HSV cone. We show successful applications of the Quaternion Split Fourier Transform and the interpolation processing. We compare these computations using the RGB metric in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {R}^{3}$ </tex-math></inline-formula> space and using the Minkowski metric in the HSV cone. This comparison shows clearly that the color image processing using the Minkowski metric in the HSV cone performs better.

Quaternion equations for hydrodynamic two-fluid model of vortex plasma

We discuss the application of quaternionic space-time algebra for the generalization of self-consistent equations describing the hydrodynamic two-fluid model of vortex plasma. It is shown that quaternionic formalism allows one to write the system of hydrodynamic equations in a compact form as one quaternion equation, which can be easy generalized to the case of damping plasma in an external electromagnetic field. As an illustration, we apply the proposed equations for the description of sound waves in electron–ion and electron–positron plasmas.

Binary Encoded Recursive Generation of Quantum Space-Times

Real geometric algebras distinguish between space and time; complex ones do not. Space-times can be classified in terms of number n of dimensions and metric signature s (number of spatial dimensions minus number of temporal dimensions). Real geometric algebras are periodic in s, but recursive in n. Recursion starts from the basis vectors of either the Euclidean plane or the Minkowskian plane. Although the two planes have different geometries, they have the same real geometric algebra. The direct product of the two planes yields Hestenes’ space-time algebra. Dimensions can be either open (for space-time) or closed (for the electroweak force). Their product yields the eight-fold way of the strong force. After eight dimensions, the pattern of real geometric algebras repeats. This yields a spontaneously expanding space-time lattice with the physics of the Standard Model at each node. Physics being the same at each node implies conservation laws by Noether’s theorem. Conservation laws are not pre-existent; rather, they are consequences of the uniformity of space-time, whose uniformity is a consequence of its recursive generation.

Real Spinors and Real Dirac Equation

We reexamine the minimal coupling procedure in the Hestenes' geometric algebra formulation of the Dirac equation, where spinors are identified with the even elements of the real Clifford algebra of spacetime. This point of view, as we argue, leads naturally to a non-Abelian generalisation of the electromagnetic gauge potential.

Electron mass singularities in semileptonic kaon decays

We show that recent improvements in the $O(\alpha)$ long-distance quantum electrodynamics (QED) corrections to the radiative inclusive $K_{e3}$ decay rate using the Sirlin representation are free from infrared divergences and collinear electron mass singularities in the limit $m_e\rightarrow0$, as predicted by the Kinoshita-Lee-Nauenberg theorem. We also verify that in massless QED with the simultaneous dimensional regularization of QED photon infrared divergences and electron mass singularities leads to the same result for the inclusive rate in the limit of four space-time dimensions. The equivalence of the two approaches results in part from an interesting interplay between a small chirality-breaking effect in the massless electron limit and the generalization of space-time algebra and phase space integrals to $d>4$ dimensions. Our finding supports the small theoretical uncertainty claimed for $K_{e3}$ radiative inclusive rates and reaffirms its utility in precision unitarity tests of the quark mixing matrix.

Generalization of Titchmarsh's theorems for the Minkowski algebra

The Minkowski space–time algebra is a powerful tool for studying the theory of electromagnetism and special relativity. In this paper, we intend to establish generalizations of Titchmarsh theorems for certain classes of functions, namely Lipschitz functions in the space for .

A signature invariant geometric algebra framework for spacetime physics and its applications in relativistic dynamics of a massive particle and gyroscopic precession

A signature invariant geometric algebra framework for spacetime physics is formulated. By following the original idea of David Hestenes in the spacetime algebra of signature (+,-,-,-), the techniques related to relative vector and spacetime split are built up in the spacetime algebra of signature (-,+,+,+). The even subalgebras of the spacetime algebras of signatures (pm ,mp ,mp ,mp ) share the same operation rules, so that they could be treated as one algebraic formalism, in which spacetime physics is described in a signature invariant form. Based on the two spacetime algebras and their “common” even subalgebra, rotor techniques on Lorentz transformation and relativistic dynamics of a massive particle in curved spacetime are constructed. A signature invariant treatment of the general Lorentz boost with velocity in an arbitrary direction and the general spatial rotation in an arbitrary plane is presented. For a massive particle, the spacetime splits of the velocity, acceleration, momentum, and force four-vectors with the normalized four-velocity of the fiducial observer, at rest in the coordinate system of the spacetime metric, are given, where the proper time of the fiducial observer is identified, and the contribution of the bivector connection is considered, and with these results, a three-dimensional analogue of Newton’s second law for this particle in curved spacetime is achieved. Finally, as a comprehensive application of the techniques constructed in this paper, a geometric algebra approach to gyroscopic precession is provided, where for a gyroscope moving in the Lense-Thirring spacetime, the precessional angular velocity of its spin is derived in a signature invariant manner.