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Geometric Algebra for Physicists
Abstract
Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with early chapters providing a self-contained introduction to geometric algebra. Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups. Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory.
... The Clifford geometric algebras can be regarded as unified language of mathematics 6,22,18 , physics 19,11,10,36 , engineering 8 , and computer science 12,7 . The Clifford geometric algebras are isomorphic to the classical matrix algebras. ...
... Let us consider the real Clifford geometric algebra , 19,20,11,32 with the identity element ≡ 1 and the generators , = 1, 2, … , , where = + ≥ 1. The generators satisfy the conditions ...
... (1) ( † ) = 0 ⇐⇒ = 0. (1) ( † ) = ⟨ † ⟩ 0 = || || 2 , where we use (24), (25), (11), and (10). Theorem 4 (Rank in GA). ...
We introduce the notion of rank of multivector in Clifford geometric algebras of arbitrary dimension without using the corresponding matrix representations and using only geometric algebra operations. We use the concepts of characteristic polynomial in geometric algebras and the method of SVD. The results can be used in various applications of geometric algebras in computer science, engineering, and physics.
... In this section we give a short introduction to Clifford algebra, loosely following the presentation of Dorst et al. [26], Dorst and De Keninck [25] and Doran and Lasenby [24]. ...
... Other algebra elements are generally denoted by upper case letters. The geometric product has to fulfill the following properties [24]: ...
... This definition only holds for vectors. For an extension to arbitrary multivectors see [24]. To further investigate the properties of these products we can look at the square of the sum of two vectors (a + b) 2 . ...
We introduce a generative model for protein backbone design utilizing geometric products and higher order message passing. In particular, we propose Clifford Frame Attention (CFA), an extension of the invariant point attention (IPA) architecture from AlphaFold2, in which the backbone residue frames and geometric features are represented in the projective geometric algebra. This enables to construct geometrically expressive messages between residues, including higher order terms, using the bilinear operations of the algebra. We evaluate our architecture by incorporating it into the framework of FrameFlow, a state-of-the-art flow matching model for protein backbone generation. The proposed model achieves high designability, diversity and novelty, while also sampling protein backbones that follow the statistical distribution of secondary structure elements found in naturally occurring proteins, a property so far only insufficiently achieved by many state-of-the-art generative models.
... As shown earlier [9][10][11][12][13][14][15][16][17][18][19][20][21], Clifford algebra formalism is fully equivalent to the traditional approach to quantum mechanics without reference to Hilbert spaces. In terms of such an approach, one can find a common basis for the description of electromagnetism and the general theory of relativity [5]. ...
... In such an algebraic structure, ideals can be obtained by multiplying an isolated element on the right or left by some elements of the ring. This leads to a simple Dirac spinor in the standard approach, but Clifford numbers contain more information about the physical properties than the representation of the spinor described in [11]. In this sense, a Clifford number is more meaningful to use to describe the origin and evolution of the universe. ...
... As demonstrated in the preceding study [9][10][11][12], quantum mechanics are a consequence of a mathematical structure that does not necessitate the introduction of an external Hilbert space of wave functions. The Dirac equation, as a transfer rule for the wave function on any manifold, has a hidden geometric structure and can be used as an interpretation for quantum mechanics [2,5,16]. ...
This article is a shortened review of previous results obtained by the author. The advantages of describing the geometric nature of the physical properties of the early universe using the Clifford algebra approach are demonstrated. A geometric representation of the wave function of the early universe is used, and a new mechanism of spontaneous symmetry breaking with different degrees of freedom is proposed. A possible supersymmetry is revealed, and it is shown that the energy of the initial vacuum can be considered equal to zero. The origin of baryonic asymmetry and the nature of dark matter can be explained using a geometric representation of the wave function of the early universe.
... In case of a spherically-symmetric and time-dependent gravitational field, it is instructive to define the components of the inverse matrix K µ α by scalar functions f 1 ≡ f 1 (t, R), g 1 ≡ g 1 (t, R), and g 2 ≡ g 2 (t, R) as follows [13]: ...
... Doran and Lasenby [13] used the tetrad formalism to formulate Einstein's field equations for the case of a massive body embedded to the FLRW universe, as a linear system of partial differential equations in the auxiliary local coordinates (t, R, θ, ϕ) for functions f 1 ≡ f 1 (t, R), g 1 ≡ g 1 (t, R), and g 2 ≡ g 2 (t, R) entering equation (17). The reader should notice that the time coordinate in the Doran-Lasenby approach remains the same as the Hubble time t. ...
... 1) The radial coordinate r is transformed to R = R(r, t) while keeping the time coordinate unchanged, t = t, in such a way that the McVittie metric (2) is transformed to the form (17) proposed by Lasenby [13]. ...
We consider the orbital motion of a test particle in the gravitational field of a massive body (that might be a black hole) with mass m placed on the expanding cosmological manifold described by the McVittie metric. We introduce the local coordinates attached to the massive body to eliminate nonphysical, coordinates-dependent effects associated with Hubble expansion. The resultant equation of motion of the test particle are analyzed by the method of osculating elements with application of time-averaging technique. We demonstrate that the orbit of the test particle is not subject to the cosmological expansion up to the terms of the second order in the Hubble parameter. However, the cosmological expansion causes the precession of the orbit of the test particle with time and changes the frequency of the mean orbital motion. We show that the direction of motion of the orbital precession depends on the Hubble parameter as well as the deceleration parameter of the universe. We give numeric estimates for the rate of the orbital precession with respect to time due to the cosmological expansion in case of several astrophysical systems.
... It describes a disk, as visualized in Figure (3). Equation (5) is a massless Weyl equation and the solution is a unit quaternion, [18,19] originating from the bivector. It spins the (2) axis either L or R, Figure (3). ...
... For this, it is essential that Alice and Bob's spins both be complex, [20,21], and carry helicity. The details are elsewhere, [5], but to obtain this equation, add the two terms in Equation (18). Note that Alice and Bob are anti-correlated, so θ will differ by π between them. ...
... In modern theoretical frameworks, Hestenes integrated quaternions into his geometric algebra formalism for quantum mechanics and relativity [34]. Doran and Lasenby [18] expanded on this, combining quaternions and geometric algebra to provide both mathematical foundations and practical applications in quantum mechanics. ...
Changing the symmetry of spin from SU(2) to the quaternion group, Q8, introduces Q-spin which has a number of ramifications. We first summarize the properties of Q-spin and then discuss some of those resulting changes. Many are counter to well-established ideas which change our view of the microscopic. Dirac’s matter-antimatter pair is challenged, being replaced by a 2D structured spin. His negative energy issue and the baryon asymmetry problem are resolved; parity is not violated in beta decay, and the existence of neutrinos is questioned. Bell’s theorem is disproven by counter example, restoring Local Realism to Nature. The singlet state is an approximation; teleportation is not feasible, quantum computing must change to remove teleportation and focus on the qubit form of Q-spin which gives deterministic outcomes of ±1. This impinges on the Foundations of Quantum Mechanics, and we discuss the reasons for the failure of Bell’s theorem; the use of Hidden Variables, wave function collapse and explain the violation of Bell’s Inequalities by showing Q-spin is a boson of odd parity in free flight, and a fermion of even parity in a polarizing field. The violation of Bell’s Inequalities is due to an anyon transition between a boson and a fermion, formulating wave-particle duality. These changes are a result of compexifying the Dirac field.
... The concept of information flow plays a key role in causal influences that are studied as an informationtheoretic setting [18,19]. Interestingly, in Ref. [20], the geometry of spacetime was shown to emerge from abstract-order lattices [21,22] expressed in a quantitative way by means of the geometric Clifford algebra language [23][24][25][26]. ...
... In three-dimensional space, ⃗ a and ⃗ b are three-dimensional vectors and this relation gives the usual results from the dot and cross products [24]. This algebra has been used to describe physical laws in different fields of physics. ...
... [1] We demonstrated that the Pythagorean theorem is derived from the fact that the interval scalars of orthogonal subspaces are additive (see Equation (10)). [2] We showed that geometric shapes are formed from the quantification of more than two equidistant coordinated chains which gives rise to multiple spatial dimensions (see Equations (24) and (33)). [3] We introduced the concept of a fence as a set of three or more collinear and coordinated chains. ...
An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in 2+1 dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras.
... He succeeded on a satisfactory physical interpretation of quantum mechanics, see for e.g. his seminal work [16] and [13,14]. ...
... with the boundary conditions (14) and the (k − 1)-vector field ∂ x · F k is uniquely determined. However, the field F k is not uniquely determined by the first condition of (14). ...
... with the boundary conditions (14) and the (k − 1)-vector field ∂ x · F k is uniquely determined. However, the field F k is not uniquely determined by the first condition of (14). If F k is a solution of the boundary value problem (14)- (16), then F k + Q k with Q k ∈ C 2 (Ω, R (k) 0,m ) such that (∂ x · Q k )| Ω = 0 and Q k | ∂Ω = 0, is another solution. ...
Clifford analysis offers suited framework for a unified treatment of higher-dimensional phenomena. This paper is concerned with boundary value problems for higher order Dirac operators, which are directly related to the Lamé-Navier and iterated Laplace operators. The conditioning of the problems upon the boundaries of the considered domains ensures their well-posedness in the sense of Hadamard.
... However, the causal interpretation, particularly in its geometric algebra (GA) version, provides profound insights into the quantum nature of spacetime and matter [17][18][19][20]. In this paper, similarly to in our previous work, we utilize spacetime algebra (STA) [20][21][22] as our mathematical language. STA is constructed from a Minkowskian vector space and provides a straightforward geometric understanding of Dirac theory. ...
... The fundamentals of GTG and its applications to spin-1 2 particles are summarized in Appendix C. It is now widely acknowledged that the quantum random motion of spin-1 2 particles can be fully described by their spin. In the presence of gravity, the spin gives rise to the torsion of spacetime [21,[25][26][27][28][29]. In particular, the effects of spin-torsion in GTG were investigated in Ref. [27]. ...
... As in our previous paper [14], we initially attempted to analyze static systems. The set ofh field that satisfies the spherically symmetric and static matter distribution is assumed to take the form [21,26]h (e t ) = f 1 e t ,h(e r ) = g 1 e r + g 2 e t , ...
The spin-torsion theory is a gauge theory approach to gravity that expands upon Einstein's general relativity (GR) by incorporating the spin of microparticles. In this study, we further develop the spin-torsion theory to examine spherically symmetric and static gravitational systems that involve free-falling macroscopic particles. We posit that the quantum spin of macroscopic matter becomes noteworthy at cosmic scales. We further assume that the Dirac spinor and Dirac equation adequately capture all essential physical characteristics of the particles and their associated processes. A crucial aspect of our approach involves substituting the constant mass in the Dirac equation with a scale function, allowing us to establish a connection between quantum effects and the scale of gravitational systems. This mechanism ensures that the quantum effect of macroscopic matter is scale-dependent and diminishes locally, a phenomenon not observed in microparticles. For any given matter density distribution, our theory predicts an additional quantum term, the quantum potential energy (QPE), within the mass expression. The QPE induces time dilation and distance contraction, and thus mimics a gravitational well. When applied to cosmology, the QPE serves as a counterpart to the cosmological constant introduced by Einstein to balance gravity in his static cosmological model. The QPE also offers a plausible explanation for the origin of Hubble redshift (traditionally attributed to the universe's expansion). The predicted luminosity distance--redshift relation aligns remarkably well with SNe Ia data from the cosmological sample of SNe Ia. In the context of galaxies, the QPE functions as the equivalent of dark matter. The predicted circular velocities align well with rotation curve data from the SPARC (Spitzer Photometry and Accurate Rotation Curves database) sample.
... However, the causal interpretation, particularly in its geometric algebra (GA) version, provides profound insights into the quantum nature of spacetime and matter [17][18][19][20]. In this paper, similarly to in our previous work, we utilize spacetime algebra (STA) [20][21][22] as our mathematical language. STA is constructed from a Minkowskian vector space and provides a straightforward geometric understanding of Dirac theory. ...
... The fundamentals of GTG and its applications to spin- 1 2 particles are summarized in Appendix C. It is now widely acknowledged that the quantum random motion of spin- 1 2 particles can be fully described by their spin. In the presence of gravity, the spin gives rise to the torsion of spacetime [21,[25][26][27][28][29]. In particular, the effects of spin-torsion in GTG were investigated in Ref. [27]. ...
... As in our previous paper [14], we initially attempted to analyze static systems. The set ofh field that satisfies the spherically symmetric and static matter distribution is assumed to take the form [21,26]h (e t ) = f 1 e t ,h(e r ) = g 1 e r + g 2 e t , ...
The spin-torsion theory is a gauge theory approach to gravity that expands upon Einstein’s general relativity (GR) by incorporating the spin of microparticles. In this study, we further develop the spin-torsion theory to examine spherically symmetric and static gravitational systems that involve free-falling macroscopic particles. We posit that the quantum spin of macroscopic matter becomes noteworthy at cosmic scales. We further assume that the Dirac spinor and Dirac equation adequately capture all essential physical characteristics of the particles and their associated processes. A crucial aspect of our approach involves substituting the constant mass in the Dirac equation with a scale function, allowing us to establish a connection between quantum effects and the scale of gravitational systems. This mechanism ensures that the quantum effect of macroscopic matter is scale-dependent and diminishes locally, a phenomenon not observed in microparticles. For any given matter density distribution, our theory predicts an additional quantum term, the quantum potential energy (QPE), within the mass expression. The QPE induces time dilation and distance contraction, and thus mimics a gravitational well. When applied to cosmology, our theory yields a static cosmological model. The QPE serves as a counterpart to the cosmological constant introduced by Einstein to balance gravity in his static cosmological model. The QPE also offers a plausible explanation for the origin of Hubble redshift (traditionally attributed to the universe’s expansion). The predicted luminosity distance–redshift relation aligns remarkably well with SNe Ia data from the cosmological sample of SNe Ia. In the context of galaxies, the QPE functions as the equivalent of dark matter. The predicted circular velocities align well with rotation curve data from the SPARC (Spitzer Photometry and Accurate Rotation Curves database) sample. Importantly, our conclusions in this paper are reached through a conventional approach, with the sole assumption of the quantum effects of macroscopic matter at large scales, without the need for additional modifications or assumptions.
... In recent years, Geometric Algebra (GA) 1 has emerged as a new alternative formalism. Originally developed by William Kingdon Clifford in 1870, it experienced a renaissance through the work of David Hestenes and the Cambridge group led by Anthony and Joan Lasenby, Chris Doran, and others [4,5,6]. ...
... Notice how e 1 ∧ e 2 plays the role of the imaginary unit i. Details for the derivation of Cauchy's theorem for integral functions can be found in [14,6,5], which we will not reproduce here. Nonetheless, we would like to highlight three points from the result: ...
... The invertibility of the ∇ function, which is Green's function, permits to invert equation (4.5) and to obtain the fields directly from the sources without passing through second-order derivatives. Meaning that Helmholtz's equation is first order, and Huygens' principle of reradiation is directly applicable and evident from the solution; see [20,6] for details. ...
Differential forms is a highly geometric formalism for physics used from field theories to General Relativity (GR) which has been a great upgrade over vector calculus with the advantages of being coordinate-free and carrying a high degree of geometrical content. In recent years, Geometric Algebra appeared claiming to be a unifying language for physics and mathematics with a high level of geometrical content. Its strength is based on the unification of the inner and outer product into a single geometric operation, and its easy interpretation. Given their similarities, in this article we compare both formalisms side-by-side to narrow the gap between them in literature. We present a direct translation including differential identities, integration theorems and various algebraic identities. As an illustrative example, we present the case of classical electrodynamics in both formalism and finish with their description of GR.
... In general, for two homogeneous multivectors Ψ α = ⟨Ψ α ⟩ α and Φ β = ⟨Φ β ⟩ β of grades α and β, respectively, the inner and outer products are defined in terms of the geometric product as [12][13][14]: ...
... Just like the relative bivectors can be expressed in terms of the relative vectors, the trivectors can be expressed as pseudo-vectors: γ α γ β γ µ = −ε αβµν Iγ ν (12) In STA, the vector derivative operator with respect to space-time position x = x µ γ µ = (t + x)γ 0 is defined as ∇ = γ µ ∂ µ = (∂ t − ∇)γ 0 (13) where x 0 = t and x = x i σ i . For brevity, we will refer to Equation (13) as the nabla operator. ...
... Just like the relative bivectors can be expressed in terms of the relative vectors, the trivectors can be expressed as pseudo-vectors: γ α γ β γ µ = −ε αβµν Iγ ν (12) In STA, the vector derivative operator with respect to space-time position x = x µ γ µ = (t + x)γ 0 is defined as ∇ = γ µ ∂ µ = (∂ t − ∇)γ 0 (13) where x 0 = t and x = x i σ i . For brevity, we will refer to Equation (13) as the nabla operator. Let us also define the structure of a generalized multivector of STA as: ...
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.
... However, the causal interpretation, particularly in its geometric algebra (GA) version, provides profound insights into the quantum nature of spacetime and matter [20,[22][23][24]. In this paper, similar to our previous work, we utilize the spacetime algebra (STA) [12,19,24] as our mathematical language. STA is constructed from a Minkowskian vector space and provides a straightforward geometric understanding of Dirac theory. ...
... It is now widely acknowledged that the quantum random motion of spin-1 2 particles can be fully described by its spin. In the presence of gravity, the spin gives rise to the torsion of spacetime [12,13,17,25,26,29]. For massive fermions, such as electrons and neutrons, the mass will always appear in the phase factor of the solutions of the wave equations. ...
... As in our previous paper [10], we attempted to analyze static systems initially. The set ofh field that satisfies the spherically-symmetric and static matter distribution is assumed to take the form [12,29]h (e t ) = f 1 e t ,h(e r ) = g 1 e r + g 2 e t , ...
The spin-torsion theory is a gauge theory approach to gravity that expands upon Einstein's general relativity (GR) by incorporating the spin of microparticles. In this study, we further develop the spin-torsion theory to examine spherically symmetric and static gravitational systems that involve free-falling macroscopic particles. We posit that the quantum spin of macroscopic matter becomes noteworthy on cosmic scales. We further assume that the Dirac spinor and Dirac equation adequately capture all essential physical characteristics of the particles and their associated processes. A crucial aspect of our approach involves substituting the constant mass in the Dirac equation with a scale function, allowing us to establish a connection between quantum effects and the scale of gravitational systems. This mechanism ensures that the quantum effect of macroscopic matter is scale-dependent and diminishes locally, a phenomenon not observed in microparticles. For any given matter density distribution, our theory predicts an additional quantum term, the quantum potential energy (QPE), within the mass expression. QPE induces time dilation, distance contraction, and thus mimics a gravitational well. When applied to cosmology, our theory yields a static cosmological model. The QPE serves as a counterpart to the cosmological constant introduced by Einstein to balance gravity in his static cosmological model. The QPE also offers a plausible explanation for the origin of the Hubble redshift (traditionally attributed to the universe's expansion). The predicted luminosity distance-redshift relation aligns remarkably well with SNe Ia data from the cosmological sample of SNe Ia. In the context of galaxies, QPE functions as the equivalent of dark matter. The predicted circular velocities align well with the rotation curve data from the SPARC (Spitzer Photometry and Accurate Rotation Curves database) sample. Importantly, our conclusions in this paper are reached through a conventional approach, with the sole assumption of the quantum effects of macroscopic matter on large scales, without the need for additional modifications or assumptions.
... Considering helical motion of electron, a theoretical explanation for the observed energy was given by Hestenes (2010). In the extensions of semi-classical theories, the spin angular momentum was identified with a bivector and the point particle executes circular motion by absorbing energy from zeropoint field (Doran & Lasenby 2003;Doran et al. 1996). Considering entirely a different kinematical approach, the extended particle structure was studied by Rivas (2002). ...
... The contravarient unit vectors are defined as 0 = 0 and = − . For elaborate details of spacetime and geometric algebra one may refer (Doran & Lasenby 2003). ...
... Combining these results with (79) gives the ground state energy 0 = ℏ 0 /2 which coincides with the ground state energy per mode of the particle oscillator in quantum description and in stochastic electrodynamics approach. In general, there may be some higher energy states of the oscillator and in such cases ≠ . ...
A new pre-quantum level physical theory underlying the quantum mechanics has been developed and its epistemological features are mainly derived from the fluctuating classical electromagnetic zeropoint field which induces random oscillations on charged particle. In complex vector space, the average random oscillations of the charged particle are treated as complex rotations so that a particle has extended structure. It is found that the quantum nature of particles arises from the superposition of internal oscillations on the center of mass motion. Using Lagrangian mechanics in complex vector space, a generalized Newton’s equation is derived and which is shown to be acceptable for both micro and macro mechanics. The generalized Newton’s equation contains additional relativistic terms at the micro level. Further, a complex Hamilton-Jacobi equation is derived and which is used to solve quantum mechanical problems in classical approach. The presence of zeropoint field also produces modifications in the central potential and the additional terms are responsible for the effects like advance of planets perihelion and deflection of light near massive objects at macro level. Further, in the case of atoms such terms lead to the estimation of Lamb shift using stochastic electrodynamics. The consideration of extended particle structure in stochastic electrodynamics allows deriving mass correction and charge correction and consequently the estimation of anomalous magnetic moment. The fundamental deeper level theory developed here, may further enhance the efforts for research connecting micro and macro aspects of matter.
... Therefore one aim of this paper is to work in this direction by looking at the effect of these three symmetries on multivectors of Cl (3,1), a (geometric) algebra that can be used to express space-time physics, withĈ,P andT symmetries, e.g., defined by Doran and Lasenby [7], but here we only focus on the effect of these transformations on the 16 basis blades that constitute Gopalan [10] and Fabrykiewicz [9] correctly turn to Clifford algebra in order to generalize the notion of cross product that only exists in three dimensions to arbitrary dimensions. In this context [17], for crystallographers it may be of interest to know that J.G. Grassmann (Justus G. was the father of Hermann G. Grassmann) originally introduced the characterization of crystal planes by orthogonal vectors, now commonly denoted with Miller indices (see Erhard Scholz [30], pp. ...
... The eight principal types S, V 0 , V, B 0 , B, T 0 , T and Q denoted in Table 1 are all uniquely characterized by the action of the elements of the group of involutions (7). It is now of course possible to follow the pattern established by Gopalan [10] and Fabrykiewicz [9] and regard linear combinations of principal types as new types, for which the action of the the group of involutions (7) would then be called mixed m. ...
... In this section we apply the symmetry operations of charge conjugationĈ, parity reversalP and time reversalT expressed in the geometric algebra Cl(3, 1) for the description of space-time as they can, e.g., be found in Doran and Lasenby [7], page 283. There the application is to spinors (even grade valued multivector functions R 1,3 → Cl + (1, 3)) including reflection at the time axis e 0 of the argument of the spinor. ...
From viewpoints of crystallography and of elementary particles, we explore symmetries of multivectors in the geometric algebra Cl(3, 1) that can be used to describe space-time.
... A universal mathematical language for physics is the so-called geometric (Clifford) algebra (GA) [1,2], a language that relies on the mathematical formalism of Clifford algebra. A very partial list of physical applications of GA methods includes the fields of gravity [3,4], classical electrodynamics [5], and massive classical electrodynamics with Dirac's magnetic monopoles [6,7]. ...
... We then extend the application of the MSTA to specify both one-qubit gates (i.e., bit-flip, phase-flip, combined bit and phase flip quantum gates, Hadamard gate, rotation gate, phase gate, and π/8-gate) and two-qubit quantum computational gates (i.e., CNOT, controlled-phase, and SWAP quantum gates) [33]. Then, employing this proposed GA description of states and gates along with the GA characterization of the Lie algebras SO(3) and SU (2) in terms of the rotor group Spin + (3, 0) formalism, we revisit from a GA perspective the proof of universality of quantum gates as discussed by Boykin and collaborators in Refs. [34,35]. ...
... In Section 4, we revisit the proof of universality of quantum gates as originally provided by Boykin and collaborators in Refs. [34,35] by making use of the material presented in Sections 2 and 3 and, in addition, by exploiting the above-mentioned GA description of the Lie algebras SO (3) and SU (2) in terms of the rotor group Spin + (3,0) formalism. We present our concluding remarks in Section 5. Finally, some technical details on the algebra of physical space cl(3) and the spacetime Clifford algebra cl(1, 3) appear in Appendix A. ...
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.
... For the interested reader, excellent introductions to geometric algebra and its calculus can be found in the texts by Hestenes & Sobczyk [7], Doran & Lasenby [8] and Dorst et al. [9]. Using the ideas of Grassman and Clifford, we can gain deep insight into the nature of the quantization of a classical system. ...
... These operations, which are highly compact, can be built up from familiar vectors and bound together by the geometric product. Following §2.6 in Doran & Lasenby [8], we first examine a simple reflection. Let n be a unit vector and a be an arbitrary vector. ...
The rotating equilibrium solutions of N identical point vortices are the stationary energy states of a higher-dimensional object projected to the x−y plane, in this case, an (N−1)-dimensional regular simplex. Parameterizing the point vortex Hamiltonian with this fundamental geometrical object leads to a simple bivector condition, which contains the equilibrium states and resolves the origin of the asymmetric solutions. This novel approach uses the geometric product and the multivector derivative, ∂ψ, of the Hamiltonian in the geometric algebra, Cl(N−1,0) to find a geometric condition that determines the equilibrium states. The resulting bivector equation is then used as the input to an optimizer, which rotates a simplex until the equilibrium condition is met, leading to a wealth of new solutions. If the vertices of the oriented simplex are projected to the x-axis, the values form the roots of the Hermite polynomials, HN(x), and obey the Stieltjes relations, capturing the collinear solutions. The vortex simplex exhibits a striking geometrical connection with the amplituhedron of quantum field theory and gives deep insight into the quantization of a classical system.
... The concept of information flow plays a key role in causal influences that are studied as an information-theoretic setting [5] [6]. Interestingly, in Ref. [12] the geometry of spacetime was shown to emerge from abstract order lattices [17] [9] expressed in a quantitative way by means of the geometric Clifford algebra language [10] [19] [7] [8]. ...
... In three dimensional space, ⃗ a and ⃗ b are three dimensional vectors and this relation gives the usual results from the dot and the cross products [10]. This algebra has been used to describe physical laws in different fields of physics. ...
Influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events form a partially-ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry, including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the Parallel postulate, the dot product, and the outer (wedge product) in 2+1 dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of geometric product in geometric Clifford algebras.
... Professor W. K. Clifford defined his geometric algebra [1] by combining and extending the Grassmann's exterior algebra [2] and Hamilton's quaternions [3] into a more general algebraic framework, which is a direct and intuitive generalization of vector algebra, with an explicit geometric interpretation [4] and clear relations with linear algebra [5,6]. Geometric algebra has developed steadily over the past century and has gained popularity by discovering many applications in different scientific fields. ...
... In the case without confusion, we can directly use 1 to replace I. By Equations (3) and (4) we have the relations between (f µ a , f a µ ) and metric as ...
In this paper we discuss the dynamic effects of the varying frames. The differential of frame or basis vectors is always equivalent to a linear transformation of the frame, and the linear transformation is not the same in different contexts. In differential geometry, the linear transformation is the connection operator. While in quantum mechanics, the operator algebra corresponds to the differentials of matrices. Corresponding to the variation of the metric, the variation of the frame contains a unusual fourth-order tensor. We also derive the Lie differential of the frame corresponding to the Lorentz transformation group. The definition of differential of the frame is different, so the corresponding linear transformation is also different. In this paper, the unified point of view to deal with the variation of frame or basis vectors will bring great convenience to the research and application of Clifford algebras.
... Clifford algebra is a universal geometric language that introduces powerful mathematical tools enabling computational efficiency in designing and manipulating geometric objects and developing new insights within a unified algebraic framework. It has been widely used in different fields of mathematics [1][2][3][4][5][6], in theoretical physics [7][8][9][10], and in computer science [11][12][13][14][15][16][17][18][19]. It was also used in many applications in image processing particularly to extend the usual definitions of Fourier transform to color images and more generally to multi-channel images; see [20]. ...
... A homography of PGL(2, C) is represented, up to the multiplication by a non-zero complex number, by a matrix of GL(2, C) a b c d (10) or equivalently, by enabling X/Y = z (considering the chart φ Y ), via the following mapping: ...
In this paper, we introduce an invariant by image viewpoint changes by applying an important theorem in conformal geometry stating that every surface of the Minkowski space R3,1 leads to an invariant by conformal transformations. For this, we identify the domain of an image to the disjoint union of horospheres ∐αHα of R3,1 by means of the powerful tools of the conformal Clifford algebras. We explain that every viewpoint change is given by a planar similarity and a perspective distortion encoded by the latitude angle of the camera. We model the perspective distortion by the point at infinity of the conformal model of the Euclidean plane described by D. Hestenesand we clarify the spinor representations of the similarities of the Euclidean plane. This leads us to represent the viewpoint changes by conformal transformations of ∐αHα for the Minkowski metric of the ambient space.
... There is a large literature on the applications of geometric algebra (see [23] for a recent survey and [8] for topic based studies), but in terms of generating understanding, there is an excellent introductory exposition in an overview of geometric algebra in [14], some informative notes in [3], two introductory text books covering algebra and calculus by Alan MacDonald [29,30]; and for a systematic exposition of geometric algebra see [7]. In mathematics geometric algebra is often called Clifford algebra after Clifford (see [10,36] for example), but Hestenes cautions against confusing geometric algebra used in physics with a Clifford algebra as a mathematical structure (see [22]). ...
... x y r cos θ sin θ − sin θ cos θ = r x cos θ − y sin θ x sin θ + y cos θ . 7 The text treats rotations and scalings, but reflections have both matrix representations and geometric algebra representations. The geometric algebra representation is particularly neat. ...
Synopsis In this article I argue that i is a quantity associated with the two-dimensional real number plane, whether as a vector, a bi-vector, a point or a transformation (rotation). This position provides a foundation for the complex numbers and accounts for complex numbers in some equations of applied mathematics and physics. I also argue that complex numbers are fundamentally geometrical and can be described by geometric algebra, and that moreover the meaning of complex numbers in physics varies with dimension and geometry of the manifold.
... [14,15]. These difficulties can be overcome by introducing the bivector, which Geometric Algebra justifies, [16]. The geometric product is the sum of the symmetric scalar product and the anti-symmetric wedge product. ...
... First, determine the larger axis, axis (X) ? <> axis (Z) (16) and then use the sign of that axis to determine the click values. Since θ varies from 0 to 2π, the magnitudes and signs of the projected axes vary. ...
{We present a statistical simulation replicating the correlation observed in EPR coincidence experiments without needing non-local connectivity. We define spin coherence as a spin attribute that complements polarization by being anti-symmetric and generating helicity. Point particle spin becomes structured with two orthogonal magnetic moments, each with a spin of —these moments couple in free flight to create a spin-1 boson. Depending on its orientation in the field, when it encounters a filter, it either decouples into two independent fermion spins of , or it remains a boson and precesses without decoupling. The only variable in this study is the angle that orients a spin on the Bloch sphere, first identified in the 1920s. There are no hidden variables. The new features introduced in this work result from changing the spin symmetry from SU(2) to the quaternion group, , which complexifies the Dirac field. The transition from a free-flight boson to a measured fermion is the reason for the observed violation of Bell's Inequalities and resolves the EPR paradox.
... Arising from Geometric Algebra [6,7], the first term describes a symmetric component that gives rise to polarization and measured Dirac spin. The second term is anti-symmetric and depends upon a bivector, iσ k , and the Levi Civita third-rank anti-symmetric tensor. ...
... The second shows the planes orthogonal to the axes: e 1 is orthogonal to e 3 Y, and e 3 is orthogonal to e 1 Y. These terms form the wedge or vector product from GA [6] leading to the formulation of helicity. In free flight, the angular momentum of the two axes, Equations (29), constructively interferes to produce the resonance spin, a boson of magnitude 1. ...
We present an analysis of the Dirac equation when the spin symmetry is changed from SU(2) to the quaternion group, Q8, achieved by multiplying one of the gamma matrices by the imaginary number, i. The reason for doing this is to introduce a bivector into the spin algebra, which complexifies the Dirac field. It then separates into two distinct and complementary spaces: one describing polarization and the other coherence. The former describes a 2D structured spin, and the latter its helicity, generated by a unit quaternion.
... Arising from Geometric Algebra, [6,7] the first term describes a symmetric component that gives rise to polarization and measured Dirac spin. The second term is anti-symmetric and depends upon a bivector, iσ k and the Levi Civita third rank anti-symmetric tensor. ...
... The second shows the planes orthogonal to the axes: e 1 is orthogonal to e 3 Y, and e 3 is orthogonal to e 1 Y. These terms form the wedge, or vector product from GA, [6] leading to the formulation of helicity. ...
We present an analysis of the Dirac equation when the spin symmetry is changed from SU(2) to the quaternion group, , afforded by multiplying one of the -matrices by the imaginary number. The reason for doing this is to introduce a bivector into the spin algebra. This complexifies the Dirac field which separates into two distinct and complementary spaces: one describing polarization and the other coherence. The former describes a 2D structured spin and the latter its helicity, generated by a unit quaternion.
... Eq.(6). This can be expressed using Geometric Algebra, [12] which for three-dimensional real space is ...
... Spin helicity is an anti-symmetric and anti-Hermitian bivector, Eq. (12), which is an element of reality of spin under quaternion symmetry, [2]. We suggest that spin is more fully defined by, σ → Σ which includes both the Pauli vector and bivector, and defines Q-spin, ...
Under the quaternion group, , spin helicity emerges as an element of reality, and complementary to spin polarization. Here we show that the correlation in EPR coincidence experiments is conserved upon separation from a singlet, being divided between polarization and coherence. Including helicity accounts for the violation of Bell's Inequalities without non-locality.
... Using a geometric algebra is one of the ways to investigate issues in modern mathematical physics [6,8,9]. We consider the real geometric algebra Cℓ 1,n . ...
It is easier to investigate phenomena in particle physics geometrically by exploring a real solution to the Dirac-Hestenes equation instead of a complex solution to the Dirac equation. The present research outlines the multidimensional Dirac-Hestenes equation. Since the matrix representation of the complexified (Clifford) geometric algebra depends on a parity of n, we explore even and odd cases separately. In the geometric algebra , there is a lemma on a unique decomposition of an element of the minimal left ideal into the product of the idempotent and an element of the real even subalgebra. The lemma is used to construct the four-dimensional Dirac-Hestenes equation. The analogous lemma is not valid in the multidimensional case, since the dimension of the real even subalgebra of is bigger than the dimension of the minimal left ideal for . Hence, we consider the auxiliary real subalgebra of to prove a similar statement. We present the multidimensional Dirac-Hestenes equation in . We prove that one might obtain a solution to the multidimensional Dirac-Hestenes equation using a solution to the multidimensional Dirac equation and vice versa. We also show that the multidimensional Dirac-Hestenes equation has gauge invariance.
... For our uncloneable encryption scheme, we make use of the structure of Clifford algebra [Lou01,DL03], widely used in quantum information theory, in particular as a generalization of the Bloch sphere representation [Die06,WW08], in operator algebras theory [Pis03,BN18], and in non-local games [Slo11,Ost16]. ...
Uncloneable encryption is a cryptographic primitive which encrypts a classical message into a quantum ciphertext, such that two quantum adversaries are limited in their capacity of being able to simultaneously decrypt, given the key and quantum side-information produced from the ciphertext. Since its initial proposal and scheme in the random oracle model by Broadbent and Lord [TQC 2020], uncloneable encryption has developed into an important primitive at the foundation of quantum uncloneability for cryptographic primitives. Despite sustained efforts, however, the question of unconditional uncloneable encryption (and in particular of the simplest case, called an uncloneable bit) has remained elusive. Here, we propose a candidate for the unconditional uncloneable bit problem, and provide strong evidence that the adversary's success probability in the related security game converges quadratically as , where K represents the number of keys and is trivially achievable. We prove this bound's validity for K ranging from 2 to 7 and demonstrate the validity up to using computations based on the NPA hierarchy. We furthemore provide compelling heuristic evidence towards the general case. In addition, we prove an asymptotic upper bound of and give a numerical upper bound of , which to our knowledge is the best-known value in the unconditional model.
... These Lie groups can be considered as generalizations of Clifford and Lipschitz groups and are important for the theory of spin groups. We hope that the explicit forms of centralizers and twisted centralizers can be useful for applications of Clifford algebras in physics [6,8,15], computer science, in particular, for neural networks and machine learning [3,4,16,21,22], image processing [2,8], and in other areas. ...
This paper investigates centralizers and twisted centralizers in degenerate and non-degenerate Clifford (geometric) algebras. We provide an explicit form of the centralizers and twisted centralizers of the subspaces of fixed grades, subspaces determined by the grade involution and the reversion, and their direct sums. The results can be useful for applications of Clifford algebras in computer science, physics, and engineering.
... The conformal model is a generalization of the homogeneous coordinate system, where 3D space is represented by a subset of Ê 5 with an inner product that no longer respects positivity. Widely used in problems of robotics, physics, and computer graphics [1,3,4], this paper applies the conformal model, for the first time (as far as we know), to represent atomic positions and calculate interatomic distances in the context of molecular geometry. As a result, we defined the C-matrix, which can perform the same function as the Z-matrix 4 , used in the calculation of the Cartesian coordinates of an atom in a molecule, in terms of its internal coordinates. ...
This paper introduces the conformal model (an extension of the homogeneous coordinate system) for molecular geometry, where 3D space is represented within R^5 with an inner product different from the usual one. This model enables efficient computation of interatomic distances using what we call the Conformal Coordinate Matrix (C-matrix). The C-matrix not only simplifies the mathematical framework but also reduces the number of operations required for distance calculations compared to traditional methods.
... and the operator describing a spin in free flight is the dyadic, σσ, see, e.g., Equation (6). This can be expressed using the well-known expression from Geometric Algebra, [24]: ...
Under the quaternion group, Q8, spin helicity emerges as a crucial element of the reality of spin and is complementary to its polarization. We show that the correlation in EPR coincidence experiments is conserved upon separation from a singlet state and distributed between its polarization and coherence. Including helicity accounts for the violation of Bell’s Inequalities without non-locality, and disproves Bell’s Theorem by a counterexample.
... Geometric Algebra Networks, also known as Clifford Algebra Networks, leverage the mathematical framework of Geometric Algebra to represent and manipulate data. Geometric Algebra is a powerful, high-dimensional algebraic system that extends traditional linear algebra, enabling the compact and intuitive representation of geometric transformations, rotations, and reflections [2,3,4]. In GA networks, data and operations are expressed in terms of multivectors, which can capture complex geometric relationships more naturally than traditional tensor or matrix representations. ...
We introduce STAResNet, a ResNet architecture in Spacetime Algebra (STA) to solve Maxwell's partial differential equations (PDEs). Recently, networks in Geometric Algebra (GA) have been demonstrated to be an asset for truly geometric machine learning. In [1], GA networks have been employed for the first time to solve partial differential equations (PDEs), demonstrating an increased accuracy over real-valued networks. In this work we solve Maxwell's PDEs both in GA and STA employing the same ResNet architecture and dataset, to discuss the impact that the choice of the right algebra has on the accuracy of GA networks. Our study on STAResNet shows how the correct geometric embedding in Clifford Networks gives a mean square error (MSE), between ground truth and estimated fields, up to 2.6 times lower than than obtained with a standard Clifford ResNet with 6 times fewer trainable parameters. STAREsNet demonstrates consistently lower MSE and higher correlation regardless of scenario. The scenarios tested are: sampling period of the dataset; presence of obstacles with either seen or unseen configurations; the number of channels in the ResNet architecture; the number of rollout steps; whether the field is in 2D or 3D space. This demonstrates how choosing the right algebra in Clifford networks is a crucial factor for more compact, accurate, descriptive and better generalising pipelines.
... Bell's Theorem justifies this [11], but how such non-local connectivity is maintained is not understood and defies rational explanation [23][24][25]. Introducing the bivector overcomes these difficulties, which Geometric Algebra justifies [26]. The geometric product is the sum of the symmetric scalar product and the antisymmetric wedge product. ...
We present a statistical simulation replicating the correlation observed in EPR coincidence experiments without needing non-local connectivity. We define spin coherence as a spin attribute that complements polarization by being anti-symmetric and generating helicity. Point particle spin becomes structured with two orthogonal magnetic moments, each with a spin of 12—these moments couple in free flight to create a spin-1 boson. Depending on its orientation in the field, when it encounters a filter, it either decouples into two independent fermion spins of 12, or it remains a boson and precedes without decoupling. The only variable in this study is the angle that orients a spin on the Bloch sphere, first identified in the 1920s. There are no hidden variables. The new features introduced in this work result from changing the spin symmetry from SU(2) to the quaternion group, Q8, which complexifies the Dirac field. The transition from a free-flight boson to a measured fermion causes the observed violation of Bell’s Inequalities and resolves the EPR paradox.
... Due to limited space, only a very small sample of geometric representations and operations made possible by the algebraic structure of GA is mentioned here. The interested reader can find much valuable information in the GA literature [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. Table 2 shows some basic multivector operations that are typically useful in practice. ...
The Geometric Algebra Fulcrum Library (GA-FuL) version 1.0 is introduced in this paper as a comprehensive computational library for geometric algebra (GA) and Clifford algebra (CA), in addition to other classical algebras. As a sophisticated software system, GA-FuL is useful for practical applications requiring numerical or symbolic prototyping, optimized code generation, and geometric visualization. A comprehensive overview of the GA-FuL design is provided, including its core design intentions, data-driven programming characteristics, and extensible layered design. The library is capable of representing and manipulating sparse multivectors of any dimension, scalar kind, or metric signature, including conformal and projective geometric algebras. Several practical and illustrative use cases of the library are provided to highlight its potential for mathematical, scientific, and engineering applications. The metaprogramming code optimization capabilities of GA-FuL are found to be unique among other software systems. This allows for the automated production of highly efficient code, based on powerful geometric modeling formulations provided by geometric algebra.
... Then, we define I ∈ Cl(E, δ),I := e 1 ∧e 2 ∧e 3 . (A.1)This element I is called the "pseudoscalar" element[10] of Cl(E, δ).Lemma A.2.In Cl(E, δ), the orthonormal set of vectors {e a } satisfy, ∀a, b ∈ Proof. By the definition of Cl(E, δ). ...
In physics, spin is often seen exclusively through the lens of its phenomenological character: as an intrinsic form of angular momentum. However, there is mounting evidence that spin fundamentally originates as a quality of geometry, not of dynamics, and recent work further suggests that the structure of non-relativistic Euclidean three-space is sufficient to define it. In this paper, we directly explicate this fundamentally non-relativistic, geometric nature of spin by constructing non-commutative algebras of position operators which subsume the structure of an arbitrary spin system. These “Spin-s Position Algebras” are defined by elementary means and from the properties of Euclidean three-space alone, and constitute a fundamentally new model for quantum mechanical systems with non-zero spin, within which neither position and spin degrees of freedom, nor position degrees of freedom within themselves, commute. This reveals that the observables of a system with spin can be described completely geometrically as tensors of oriented planar elements, and that the presence of non-zero spin in a system naturally generates a non-commutative geometry within it. We will also discuss the potential for the Spin-s Position Algebras to form the foundation for a generalisation to arbitrary spin of the Clifford and Duffin–Kemmer–Petiau algebras.
... . The interested reader can refer to the works from Suter (Suter, 2003), Doran and Lasenby (Doran & Lasenby, 2003), and Miller (Miller, 2013) for a deeper introduction to Geometric Algebra. ...
Ever since AC devices and systems achieved importance and later dominance in production and transmission of electric energy, engineers have been interested in quantifying their performance. In the general (polyphase, nonsinusoidal) case, the problem is made more difficult by possible presence of couplings among phases and of higher harmonics (due to sources and to circuit nonlinearities).
We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions D>2 and give an exact solution of the oscillator equations in the general case. At the supercritical Hopf bifurcation marked by a transition from stable fixed-point dynamics to oscillatory motion, the Jacobian matrix evaluated at the fixed point has N=⌊D/2⌋ pairs of complex conjugate eigenvalues which cross the imaginary axis simultaneously. For odd D there is an additional purely real eigenvalue that does the same. Oscillatory dynamics is asymptotically confined to a hypersphere SD−1 and is characterised by extreme multistability, namely the coexistence of an infinite number of limiting orbits, each of which has the geometry of a torus TN on which the motion is either periodic or quasiperiodic. We also comment on similar Clifford extensions of other limit cycle oscillator systems and their generalizations.
好的问题是扩展科学疆域的起点和阶梯。本文汇编了作者在研究统一场论过程中遇到了但没有解决好的几个问题，整理出来向诸位同仁请教. 这些问题包括标架与度规的代数与变分关系，超复数的零模集的意义，非线性旋量场的负能禁闭以及与Pauli不相容原理的关系，奇点定理的反例与动力学分析等，都具有明确的物理意义。本文简略地介绍了问题的背景材料和已有的一些进展，以便于读者快速理解问题的本质和现状。这些问题的解决，可能会提升人们对大自然深层规律的认识。
A good problem is the starting point and the ladder of expanding the territory of science. This paper collects several problems that the author encountered in the process of studying the unified field theory but did not solve well, and sorts them out to consult colleagues in the field. These problems include the algebraic and variational relations between frames and metric, the significance of the zero norm set of hypercomplex numbers, the negative energy confinement of nonlinear spinor field and the relationship with Pauli exclusion principle, the counterexamples and dynamic analysis of the singularity theorem, which all have clear physical significance. This paper briefly introduces the background material of the problems and the existing progress, so that the readers can quickly understand the nature and current status of the problems. The solution of these problems may enhance people's understanding of the deep laws of nature.
Keywords: Hypercomplex numbers; Clifford algebras; Nonlinear spinors; Singularity theorems; Unified field theory
MSC 2020: 15A66, 15A67, 16W55, 83-08
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.
For many applications, it can be useful to consider multivectorial algebra, taking benefits of multidimensional tools, especially for multiphysical applications. It can be applied for problems of fluid mechanics or solid mechanics. It can also be useful for electrodynamics, as well as the couplings between these domains. For the latters, multivectorial approaches are generally disregarded for different reasons (simplicity, ignorance...), whereas the use of such a more advanced algebra can present interesting properties for modelling of physical and mechanical systems. The aim of this article is to present applications of Clifford algebra to model the motion of a chain of magnets in a fluid flow, leading to electromagnetism induction in a flat coil, that can be used as a sensor. Reversible as well as irreversible phenomena can thus be considered by use of biquaternions to describe such a case.
The dynamics of the torsion field is analyzed in the framework of the Covariant Canonical Gauge Theory of Gravity (CCGG), a De Donder–Weyl Hamiltonian formulation of gauge gravity. The action is quadratic in both, the torsion and the Riemann–Cartan tensor. Since the latter adds the derivative of torsion to the equations of motion, torsion is no longer identical to spin density, as in the Einstein–Cartan theory, but an additional propagating degree of freedom. As torsion turns out to be totally anti-symmetric, it can be parametrised via a single axial vector. It is shown in this paper that, in the weak torsion limit, the axial vector obeys a wave equation with an effective mass term which is partially dependent on the scalar curvature. The source of torsion is thereby given by the fermion axial current which is the net fermionic spin density of the system. Possible measurable effects and approaches to experimental analysis are addressed. For example, neutron star mergers could act as a dipoles or quadrupoles for torsional radiation, and an analysis of radiation of pulsars could lead to a detection of torsion wave background radiation.
In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics.
It is shown that every Lie algebra can be represented as a bivector alge- bra; hence every Lie group can be represented as a spin group. Thus, the computa- tional power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thor- oughly analyzed, and an invariant method for constructing real spin representations of other classical groups is developed. Moreover, it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra.
A tutorial of geometric calculus is presented as a continuation of the development of geometric algebra in a previous paper. The geometric derivative is defined in a natural way that maintains the close correspondence between geometric algebra and the algebra of real numbers. The use of geometric calculus in physics is illustrated by expressing some basic results of electrodynamics.
Spacetime algebra is employed to formulate classical relativis- tic mechanics without coordinates. Observers are treated on the same footing as other physical systems. The kinematics of a rigid body are expressed in spinor form and the Thomas precession is derived.
This paper presents a tutorial of geometric algebra, a very useful but generally unappreciated extension of vector algebra. The emphasis is on physical interpretation of the algebra and motives for developing this extension, and not on mathematical rigor. The description of rotations is developed and compared with descriptions using vector and matrix algebra. The use of geometric algebra in physics is illustrated by solving an elementary problem in classical mechanics, the motion of a freely spinning axially symmetric rigid body.
The Pauli theory of electrons is formulated in the language of multivector calculus. The advantages of this approach are demonstrated in an analysis of local observables. Planck's constant is shown to enter the theory only through the magnitude of the spin. Further, it is shown that, when obtained as a limiting case of the Pauli theory, the Schrödinger theory describes a particle with constant local spin. An important consequence of this result is the realization that the usual interpretations of the Dirac and Schrödinger theories are mutually inconsistent in certain details.
A spinor formulation of the classical Lorentz force is given which describes the presession of an electron's spin as well as its velocity. Solutions are worked out applicable to an electron in a
The spin-torsion sector of a new gauge-theoretic formulation of gravity is analysed and the relationship to the Einstein-Cartan-Kibble-Sciama theory of gravity is discussed. The symmetries of the Riemann tensor and the con-servation laws of the theory are derived. This formalism is applied to the problem of a Dirac field coupled self-consistently to gravity. The equations derived from a minimally-coupled gauge-invariant Lagrangian naturally give the gauge-theoretic analogues of the Einstein-Cartan-Dirac equations. Finally, a semi-classical model for a spinning point-particle moving in a gravitational background with torsion is considered.
This paper employs the ideas of geometric algebra to investigate the physical content of Dirac's electron theory. The basis is Hestenes' discovery of the geometric significance of the Dirac spinor, which now represents a Lorentz transformation in spacetime. This transformation specifies a definite velocity, which might be interpreted as that of a real electron. Taken literally, this velocity yields predictions of tunnelling times through potential barriers, and defines streamlines in spacetime that would correspond to electron paths. We also present a general, first-order diffraction theory for electromagnetic and Dirac waves. We conclude with a critical appraisal of the Dirac theory.
Projective geometry is formulated in the language of geometric algebra, a unified mathematical language based on Clifford algebra. This closes the gap between algebraic and synthetic approaches to projective geometry and facilitates connections with the rest of mathematics.
As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18)
Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with a simple projective interpretation. This opens up new possibilities for coordinate-free computations in linear algebra. For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar. This approach also reveals deep relations between the structure of the linear geometries, from projective to metrical, and the structure of Clifford algebras. This is apparent in a new relation between additive and multiplicative forms for intervals in the cross-ratio. Also, various factorizations of Clifford algebras into Clifford algebras of lower dimension are shown to have projective interpretations.As an important application with many uses in physics as well as in mathematics, the various representations of the conformal group in Clifford algebra are worked out in great detail. A new primitive generator of the conformal group is identified.
A new method for calculating the curvature tensor is developed and applied to the Scharzschild case. The method employs Clifford algebra and has definite advantages over conventional methods using differential forms or tensor analysis.
A new calculus, based upon the multivector derivative, is developed for Lagrangian mechanics and field theory, providing streamlined and rigorous derivations of the Euler-Lagrange equations. A more general form of Noether's theorem is found which is appropriate to both discrete and continuous symmetries. This is used to find the conjugate currents of the Dirac theory, where it improves on techniques previously used for analyses of local observables. General formulas for the canonical stress-energy and angular-momentum tensors are derived, with spinors and vectors treated in a unified way. It is demonstrated that the antisymmetric terms in the stress-energy tensor are crucial to the correct treatment of angular momentum. The multivector derivative is extended to provide a functional calculus for linear functions which is more compact and more powerful than previous formalisms. This is demonstrated in a reformulation of the functional derivative with respect to the metric, which is then used to recover the full canonical stress-energy tensor. Unlike conventional formalisms, which result in a symmetric stress-energy tensor, our reformulation retains the potentially important antisymmetric contribution.
This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a geometric product of vectors in 2-and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analyzed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics), Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained—results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics.
The spacetime algebra (STA) is the natural, representation-free language for Dirac's theory of the electron. Conventional Pauli, Dirac, Weyl, and Majorana spinors are replaced by spacetime multivectors, and the quantum - and -matrices are replaced by two-sided multivector operations. The STA is defined over the reals, and the role of the scalar unit imaginary of quantum mechanics is played by a fixed spacetime bivector. The extension to multiparticle systems involves a separate copy of the STA for each particle, and it is shown that the standard unit imaginary induces correlations between these particle spaces. In the STA, spinors and operators can be manipulated without introducing any matrix representation or coordinate system. Furthermore, the formalism provides simple expressions for the spinor bilinear covariants which dispense with the need for the Fierz identities. A reduction to2+1 dimensions is given, and applications beyond the Dirac theory are discussed.
Using the spacetime algebra formulation of the Dirac equation, we demonstrate how to perform cross-section calculations following a method suggested by Hestenes (1982). Instead of an S-matrix, we use an operator that rotates the initial states into the scattered states. By allowing the scattering operator to become a function of the initial spin, we can neatly handle spin-dependent calculations. When the operator is independent of spin, we can provide manifestly spin-independent results. We use neither spin basis nor spin sums, instead handling the spin orientation directly. As examples, we perform spin-dependent calculations in Coulomb scattering to second order, and briefly consider more complicated calculations in QED.
We investigate the infinite, straight, rotating cosmic string within the framework of Einstein's general theory of relativity. A class of exact interior solutions is derived for which the source satisfies the weak and the dominant energy conditions. The interior metric is matched smoothly to the exterior vacuum. A subclass of these solutions has closed timelike curves both in the interior and the exterior geometries.
A new gauge-theoretic approach to gravity is applied to the study of
rotating cylindrically symmetric strings. The interior and exterior
equations are reduced to a simple set of first-order differential
equations, and suitable matching conditions are obtained. The gauge
theory formulation affords a clear understanding of the physical
observables in the theory, and provides simple conditions for the
properties of the fields on the string axis. In this context some errors
in previously published work are exposed. Three situations are
discussed: the vacuum region; pressure-free matter; and a
(2+1)-dimensional ``ideal fluid.'' In each case a set of analytic
solutions is presented. It is shown that if the fluid is rotating
rigidly then closed timelike curves are inevitable, despite the fact
that the matter satisfies the weak energy condition.
This paper develops a geometric model for coupled two-state quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric algebra of a Euclidean vector space. This algebra is then lifted to Minkowski space-time and its associated geometric algebra, and the insights this provides into how density operators and entanglement behave under Lorentz transformations are discussed. The direct sum of multiple copies of space-time induces a tensor product structure on the associated algebra, in which a suitable quotient is isomorphic to the matrix algebra conventionally used in multi-qubit quantum mechanics. Finally, the utility of geometric algebra in understanding both unitary and nonunitary quantum operations is demonstrated on several examples of interest in quantum information processing.
We consider topological contributions to the action integral in a gauge theory formulation of gravity. Two topological invariants are found and are shown to arise from the scalar and pseudoscalar parts of a single integral. Neither of these action integrals contribute to the classical field equations. An identity is found for the invariants that is valid for non-symmetric Riemann tensors, generalizing the usual GR expression for the topological invariants. The link with Yang-Mills instantons in Euclidean gravity is also explored. Ten independent quadratic terms are constructed from the Riemann tensor, and the topological invariants reduce these to eight possible independent terms for a quadratic Lagrangian. The resulting field equations for the parity non-violating terms are presented. Our derivations of these results are considerably simpler that those found in the literature.
A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the positions and orientations of the matter fields. In this manner all properties of the background spacetime are removed from physics, and what remains are a set of `intrinsic' relations between physical fields. The properties of the gravitational gauge fields are derived from both classical and quantum viewpoints. Field equations are then derived from an action principle, and consistency with the minimal coupling procedure selects an action that is unique up to the possible inclusion of a cosmological constant. This in turn singles out a unique form of spin-torsion interaction. A new method for solving the field equations is outlined and applied to the case of a time-dependent, spherically-symmetric perfect fluid. A gauge is found which reduces the physics to a set of essentially Newtonian equations. These equations are then applied to the study of cosmology, and to the formation and properties of black holes. The existence of global solutions enables one to discuss the properties of field lines inside the horizon due to a point charge held outside it. The Dirac equation is studied in a black hole background and provides a quick derivation of the Hawking temperature. Comment: 112 pages, 6 figures. Published in Phil. Trans. R. Soc. Lond. A 356, 487-582 (1998). Revised version with some corrections and improvements
The differential cross section for scattering of a Dirac particle in a black hole background is found. The result is the gravitational analog of the Mott formula for scattering in a Coulomb background. The equivalence principle is neatly embodied in the cross section, which depends only on the incident velocity, and not the particle mass. The low angle limit agrees with classical calculations based on the geodesic equation. The calculation employs a well-defined iterative scheme which can be extended to higher orders. Repeating the calculation in different gauges shows that our result for the cross section is gauge-invariant and highlights the issues involved in setting up a sensible iterative scheme. Comment: 11 latex pages, no figures
The problem of spacecraft orientation through observations of space objects having zero as well as nonzero daily parallax is solved by means of quaternion algebra. It is assumed that there are unit vectors, which are known directions of several space objects in the inertial reference system, and the same unit vectors are measured in the body-fixed reference system. A new quaternion algorithm is suggested for the case when measurements of 'near-body' objects are used. The solution of the problem is formulated as a quaternion conditional extremum problem, and both the position vector and three-axis attitude of the spacecraft are determined by the Lagrange multipliers method. The application of quaternions avoids numerous trigonometric calculations and gives a solution suitable for both theoretical analysis and computations. The method can also be useful in space photogrammetry and astronomy.
We present a simple derivation of the Schwarzschild and Kerr geometries by simplifying the Einstein free space field equations for the algebraically special form of metric studied by Kerr. This results in a system of two partial differential equations, the Laplace and eikonal equations, for a complex generating function. The metric tensor is a simple explicit functional of this generating function. The simplest solution generates the Schwarzschild geometry, while a displacement of the origin by ia in this solution generates the Kerr geometry.
The authors evaluate the differential cross section for scattering from a Schwarzschild source, the analogue of Rutherford scattering. The classical formula is compared with the Born approximation.
A semirelativistic two‐body Dirac equation with an enlarged set of phenomenological potentials, including Breit‐type terms, is investigated for the general case of unequal masses. Solutions corresponding to definite total angular momentum and parity are shown to fall into two classes, each one being obtained by solving a system of four coupled first‐order radial differential equations. The reduction of each of these systems to a pair of coupled Schrödinger‐type equations is also discussed.
The general problem of scattering and absorption of waves from a Schwarzschild black hole is investigated. A scattering absorption amplitude is introduced. The unitarity theorem for this problem is derived from the wave equation and its boundary conditions. The formulation of the problem, within the formal scattering theory approach, is also given. The existence of a singularity in space-time is related explicitly to the presence of a nonzero absorption cross section. Another derivation of the unitarity theorem for our problem is given by operator methods. The reciprocity relation is also proved; that is, for the scattering of waves the black hole is a reciprocal system. Finally, the elastic scattering problem is considered, and the elastic scattering amplitude is calculated for high frequencies and small scattering angles.
The properties of a Schwarzschild black hole as an elastic scatterer of waves are studied in detail. Phase shifts and elastic scattering cross sections are obtained for a wide range of energy and angular momentum from the exact analytical solutions of the wave equation by employing computational techniques developed by the author in an earlier paper. Features of the low-frequency elastic scattering are pointed out, and comparison with other approximations is made. The angular distribution of the scattered waves is obtained for different values of the frequency. As could be expected, it presents a peak (∝θ-4) in the forward direction and a "glory" in the backward direction. For intermediate angles it shows a complicated oscillatory behavior as frequency increases that disappears only at the geometrical-optics limit. By using the elastic scattering parameters obtained here, we calculate the angular distribution of the absorbed waves. The differential absorption cross section is isotropic for low frequencies and gradually shows features of a diffraction pattern as frequency increases. It shows an absolute maximum in the forward direction which grows and narrows with the frequency. In the geometrical-optics limit there results a Dirac δ function.
A reformulation of Grassmann calculus is presented in terms of geometric algebra—a unified language for physics based on Clifford algebra. In this reformulation, Grassmann generators are replaced by vectors, so that every product of generators has a natural geometric interpretation. The calculus introduced by Berezin [The Method of Second Quantization (Academic, New York, 1966)] is shown to be unnecessary, amounting to no more than an algebraic contraction. This approach is not only conceptually clearer, but it is also computationally more efficient, as demonstrated by treatments of the ‘‘Grauss’’ integral and the Grassmann Fourier Transform. The reformulation is applied to pseudoclassical mechanics [Ann. Phys. 104, 336 (1977)], where it is shown to lead to a new concept, the multivector Lagrangian. To illustrate this idea, the three‐dimensional Fermi oscillator is reformulated and solved, and its symmetry properties discussed. As a result, a new and highly compact formula for generating super‐Lie algebras is revealed. The paper ends with a discussion of quantization, outlining a new approach to fermionic path integrals.
It is argued that geometric algebra, in the form of the multiparticle spacetime algebra, is well suited to the study of multiparticle quantum theory, with advantages over conventional techniques both in ease of calculation and in providing an intuitive geometric understanding of the results. This is illustrated by comparing the geometric algebra approach for a system of two spin-1/2 particles with the nonrelativistic approach of Holland [Phys. Rep. 169, 294 (1988)]. © 1999 American Institute of Physics.
We develop cosmological theory from first principles starting with curvature coordinates (R,T) in terms of which the metric has the form ds2(R,T)=dR2/A(R,T)+R2dΩ2-B(R,T)dT2. The Einstein field equations, including cosmological constant, are given for arbitrary Tνμ, and the timelike geodesic equations are solved for radial motion. We then show how to replace T with a new time coordinate τ that is equal to the time measured by radially moving geodesic clocks. Cosmology is brought into the picture by setting Tνμ equal to the stress-energy tensor for a perfect fluid composed of geodesic particles, and letting τ be the time measured by clocks coincident with the fluid particles. We solve the field equations in terms of (R,τ) coordinates to get the metric coefficients in terms of the pressure and density of the fluid. The metric on the subspace τ=const is equal to dR2+R2dΩ2, and so is flat, with R having the physical significance that it is a measure of proper distance in this subspace. As specific examples, we consider the de Sitter and Einstein—de Sitter universes. On an (R,τ) spacetime diagram, all trajectories in an Einstein—de Sitter universe are emitted from R=0 at the "big bang" at τ=0. Further, a light signal coming toward R=0 at some time τ>0 will, in its past history, have started from R=0 at τ=0, and have turned around on the line 2R=3τ. A consequence of this is a "tilting" of the null cones along the trajectory of a cosmological particle. The turnaround line 2R=3τ marks the transition where an R=const line changes from spacelike to timelike in character. We show how to apply the techniques developed here to the inhomogeneous problem of a Schwarzschild mass imbedded in a given universe in the paper immediately following this one.
A new technique for solving the perturbed Kepler problem is presented. As its approach is essentially algebraic, it can be easily computerized and carried out to any perturbation order. It also eliminates the notorious difficulty of small divisors, and is well suited to deal with commensurable orbital periods, as demonstrated by its explanation of Kirkwood gaps.
In this paper, we study hyperbolic conformal geometry following a Clifford algebraic approach. Similar to embedding an affine space into a one-dimensional higher linear space, we embed the hyperboloid model of the hyperbolic n-space in
Rn,1{\mathcal{R}}^{n,1}
into
Rn + 1,1{\mathcal{R}}^{n + 1,1}
. The model is convenient for the study of hyperbolic conformal properties. Besides investigating various properties of the model, we also study conformal transformations using their versor representations.
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.
The generator of electromagnetic gauge transformations in the Dirac equation has a unique geometric interpretation and a unique extension to the generators of the gauge group SU(2) U(1) for the Weinberg-Salam theory of weak and electromagnetic interactions. It follows that internal symmetries of the weak interactions can be interpreted as space-time symmetries of spinor fields in the Dirac algebra. The possibilities for interpreting strong interaction symmetries in a similar way are highly restricted.
Invariant methods for formulating and analyzing the mechanics of the skeleto-muscular system with geometric algebra are further developed and applied to reaching kinematics. This work is set in the context of a neurogeometry research program to develop a coherent mathematical theory of neural sensory-motor control systems.
A new invariant formulation of 3D eye-head kinematics improves on the computational advantages of quaternions. This includes a new formulation of Listing's Law parametrized by gaze direction leading to an additive, rather than a multiplicative, saccadic error correction with a gaze vector difference control variable. A completely general formulation of compensatory kinematics characterizes arbitrary rotational and translational motions, vergence computation, and smooth pursuit. The result is an invariant, quantitative formulation of the computational tasks that must be performed by the oculomotor system for accurate 3D gaze control. Some implications for neural network modeling are discussed.
An objective account of the action of a Stern-Gerlach apparatus on particles is given, using the Dirac equation. This generalizes earlier work on a causal interpretation of the Pauli equation to the relativistic domain, leading to a more natural choice for the current in the model.
We show how the basic operations of quantum computing can be expressed and manipulated in a clear and concise fashion using a multiparticle version of geometric (aka Clifford) algebra. This algebra encompasses the product operator formalism of NMR spectroscopy, and hence its notation leads directly to implementations of these operations via NMR pulse sequences.
Spacetime algebra (STA) is the name given to the geometric (Clifford) algebra where vectors are equipped with a product that is associative and distributive over addition. The essential feature of this product is that it mixes two different types of object such as scalars and bivectors. There is a growing realization that geometric algebra provides a unified and powerful tool for the study of many areas of mathematics, physics, and engineering. The only impediment to the wider adoption of geometric algebra appears to be physicists' understandable reluctance to adopt new techniques. Others include following the streamlines for two particles through a scattering event, or using the 3-particle algebra to model pair creation. It will also be of considerable interest to develop simplified techniques for handling more complicated many-body problems. The STA is a powerful tool for classical relativistic physics.
We analyse a supersymmetric mechanical model derived from (1+1)-dimensional field theory with Yukawa interaction, assuming that all physical variables take their values in a Grassmann algebra B. Utilizing the symmetries of the model we demonstrate how for a certain class of potentials the equations of motion can be solved completely for any B. In a second approach we suppose that the Grassmann algebra is finitely generated, decompose the dynamical variables into real components and devise a layer-by-layer strategy to solve the equations of motion for arbitrary potential. We examine the possible types of motion for both bosonic and fermionic quantities and show how symmetries relate the former to the latter in a geometrical way. In particular, we investigate oscillatory motion, applying results of Floquet theory, in order to elucidate the role that energy variations of the lower order quantities play in determining the quantities of higher order in B. Comment: 29 pages, 2 figures, submitted to Annals of Physics