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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.
... (2 √ 2πGℏ 3 √‖̂‖ ∇∇ − √Λ ) −̂ℏ = 0 (1) ...
... If you do not know what I am talking about, I strongly recommend you check the masterpiece [1] and the best collection of Geometric Algebra knowledge [3]. ...
... The ̃ could be the be the reverse of (being the reverse commented in [3]), or any other operation that affects the sign of the certain components of the wavefunction multivector (or reverse the order of its basis vectors). Or it could be exactly the same . ...
In this paper, we will use Geometric Algebra to combine the Einstein Field Equations with the Dirac Equation, arriving to the following equation: + (1 + 1 √2‖̂‖ ‖ ‖) −1 (2 √ 2πGℏ 3 √‖̂‖ ∇∇ − √Λ) −̂ℏ = 0 (1) Where ψ is the wavefunction o the particle and g 1/2 is the collection of the basis vectors which square is the collection of the product of the basis vectors by themselves (the metric g in Geometric Algebra representation). And the t represents the trivector in Geometric Algebra Cl3,0. The original aim of the paper was to demonstrate that the muon g-2 discrepancy could be explained by gravitational effects created by its own muon mass (not even necessary to consider Earth's or Sun's gravity. The difference of muon g-2 between measured value and the theoretical value is: = − ℎ = 2.89 − 9 (9.4) Solving equation (1) is very complicated and depends on the boundary conditions. But the set of the possible solutions will depend only in its coefficients and the boundary conditions. Applying only the coefficients of equation (1) and as a possible boundary condition the classical radius of the muon, we can get solutions as: J. Sánchez 2 1 = √ 2√ 2πGℏ 3 √ 1 2 + (1 √2) 2 = 2.78 − 9 (9.7) That has less than 4% error compared to the expected value. − 1 = 3.95% (9.8) There are other possibilities, as: 2 = 4 √ 2 √ 2πGℏ 3 ℏ = 2.617 − 9 (9.9) With a higher error: − 2 = 9.44% (9.10) None of these solutions gives the exact value, but it has been demonstrated that yes, the muon g-2 discrepancy could be due to gravitational effects (they are in the order of the difference between the measured value and the theoretical value). It is important to remark that the gravitational effects in this regard are due the mass of the muon itself, it is not necessary to include higher gravitational potentials as the Earth or the Sun gravity. Just the muon mass is capable of creating this discrepancy. Also, we have seen how using Geometric algebra it is possible to combine gravitational equations (generally used in tensor formalism) with the Dirac Equation (normally used in Matrix formalism).
... R) when dealing with 3D Euclidean geometry. For Geometric Algebra (GA), on the other hand, we will stick to the notation commonly employed in the field [26,27], hence we will use simple lowercase letters for vectors (e.g. e 1 , e 2 , a, t, ...), uppercase letters for elements with grade 2 or higher (e.g. rotors R, T a , M , ...) or elements in CGA (e.g. ...
... Starting from the second half of the last century, GA has found application in many fields, including physics [26,35], computer vision [36,37,38], graphics [39,40] and molecular modelling [41,42,43]. ...
... A rotor is equivalent to a quaternion in 3D, it has fewer parameters than a rotation matrix, it does not suffer from gimbal lock and can be extended to any arbitrary dimension. We will show how translations can also be represented via rotors in Section 3. We refer the reader to [26] for a more complete discussion of GA fundamentals. ...
We introduce CGA-PoseNet, which uses the 1D-Up approach to Conformal Geometric Algebra (CGA) to represent rotations and translations with a single mathematical object, the motor, for camera pose regression. We do so starting from PoseNet, which successfully predicts camera poses from small datasets of RGB frames. State-of-the-art methods, however, require expensive tuning to balance the orientational and translational components of the camera pose.This is usually done through complex, ad-hoc loss function to be minimized, and in some cases also requires 3D points as well as images. Our approach has the advantage of unifying the camera position and orientation through the motor. Consequently, the network searches for a single object which lives in a well-behaved 4D space with a Euclidean signature. This means that we can address the case of image-only datasets and work efficiently with a simple loss function, namely the mean squared error (MSE) between the predicted and ground truth motors. We show that it is possible to achieve high accuracy camera pose regression with a significantly simpler problem formulation. This 1D-Up approach to CGA can be employed to overcome the dichotomy between translational and orientational components in camera pose regression in a compact and elegant way.
... A major theorem, first proved by Brauer & Weyl (1935) [12], is that the algebra of outer products of spinors is isomorphic to the Clifford algebra [13,14,15] of multivectors, (see [16] for a history), also known as the geometric algebra [17,18,19,20], in any number of spacetime dimensions. ...
... Notwithstanding the equality of ε and k γ + k (or of ε alt and k iγ − k ) in the chiral representation, ε (or ε alt ) is defined to transform as a Lorentzinvariant spinor tensor under rotations, not as an element of the geometric algebra. The vectors γ + k and γ − k in the expressions (19) and (20) are to be interpreted as being spacelike (timelike vectors being treated as i times a spacelike vector). The spinor metric ε or ε alt is the same independent of whether orthonormal vectors are spacelike or timelike. ...
... The spinor metric ε, in either of the forms (19) or (20), is real and orthogonal, and its square is plus or minus the unit matrix, ...
Spinors are central to physics: all matter (fermions) is made of spinors, and all forces arise from symmetries of spinors. It is common to consider the geometric (Clifford) algebra as the fundamental edifice from which spinors emerge. This paper advocates the alternative view that spinors are more fundamental than the geometric algebra. The algebra consisting of linear combinations of scalars, column spinors, row spinors, multivectors, and their various products, can be termed the supergeometric algebra. The inner product of a row spinor with a column spinor yields a scalar, while the outer product of a column spinor with a row spinor yields a multivector, in accordance with the Brauer-Weyl (1935) theorem. Prohibiting the product of a row spinor with a row spinor, or a column spinor with a column spinor, reproduces the exclusion principle. The fact that the index of a spinor is a bitcode is highlighted.
... Independently from the above studies, the mathematician G. Sobczyk has conducted systematic studies on the representation of the conventional Pauli matrices, via geometric Clifford algebras (see Refs. [32][33][34][35][36] and references quoted therein). ...
... Thus, the so called Pauli vectors e k ≡ e k , k = 1, 2, 3 , are identified with the unit vectors e k along the xyz-axes (for details, see Refs. [32][33][34][35][36] . ...
... By upgrading and extending Ref. 31 , we now study the representation of nuclear magnetic moments via the reformulation of iso-Pauli matrices (18) 16 in terms of geometric algebra (21) 33 . For this purpose, we note that when formulated on the associative envelope ξ , the iso-Pauli matrices satisfy all algebraic properties of the conventional Pauli matrices. ...
In this paper, we outline the research conducted by the first named author and his associates on the axiom-preserving, thus isotopic completion of quantum mechanics into hadronic mechanics according to the historical legacy by A. Einstein, B. Podolsky and N. Rosen that quantum mechanics is not a complete theory and review the ensuing exact representation of the magnetic moment and spin of the Deuteron in its ground state thanks to the isotopic completion of Pauli’s matrices with an explicit and concrete content of D. Bohm’s hidden variableλ. We then outline the independent studies conducted by the second named author on the representation of the conventional Pauli’s matrices via geometric Clifford algebras. We finally show that the combination of the two studies allows a mathematically rigorous, numerically exact and time invariant geometric representation of the magnetic moment, spin and hidden variable of the Deuteron in its ground state.
... In all the above equations I(·) expresses the inertia tensor being applied to a multivector or to each multivector element in the matrix case. The inertia tensor is a grade-preserving operation since it maps bivectors to bivectors [54]. ...
... Similarly to the previous section, the inverse kinematics cost function can be used to define target poses for a manipulator to reach. More interesting is to consider Equation (54) for manipulators, of course in this case the motor again corresponds to the forward kinematics function. Using X = e 0 therefore means that the expression M (q)X M (q) corresponds to the tip of the end-effector. ...
... 1) Reaching a Point: Reaching a point means that the desired geometric primitive is a point, i.e. X d = P in Equation (54). The reference primitive X r is a point as well and represents the tip of the end-effector. ...
Many problems in robotics are fundamentally problems of geometry, which lead to an increased research effort in geometric methods for robotics in recent years. The results were algorithms using the various frameworks of screw theory, Lie algebra and dual quaternions. A unification and generalization of these popular formalisms can be found in geometric algebra. The aim of this paper is to showcase the capabilities of geometric algebra when applied to robot manipulation tasks. In particular the modelling of cost functions for optimal control can be done uniformly across different geometric primitives leading to a low symbolic complexity of the resulting expressions and a geometric intuitiveness. We demonstrate the usefulness, simplicity and computational efficiency of geometric algebra in several experiments using a Franka Emika robot. The presented algorithms were implemented in c++20 and resulted in the publicly available library \textit{gafro}. The benchmark shows faster computation of the kinematics than state-of-the-art robotics libraries.
... The approach to Majorana degrees of freedom taken in this thesis is conceptually similar to the geometric algebra framework of mathematical physics [166][167][168]. In contrast to standard vector calculus in physics where vectors are taken as elements living in sections of the tangent bundle, in geometric algebra vectors representing physical quantities are taken to be elements of sections of a Clifford bundle. ...
... For example, the transformations in [162] were based on realizing representations of SU(2) S ⊗ SU(2) I , or more generally, SU(2 n ) ⊗ SU(2 n ). The approach taken here is a more abstract and less model dependent approach and is conceptually similar to, and motivated by, the geometric algebra framework of mathematical physics [166][167][168]. As a primer for the generalized Majorana formalism developed below, presented here is a brief review of geometric algebra. ...
... An orthonormal basis for the real k-dimensional Clifford algebra Cl k (R) is given by {e n } such that a Clifford vector a can be written in components as a = a 1 e 1 + · · · + a n e n . The product of two Clifford vectors a and b is described by [166][167][168] which are termed the scalar, vector, bivector, and trivector basis elements respectively. These vectors are bases of contravariant vector spaces. ...
Presented in this thesis are a set of projects which lie at the intersection between strong correlations and topological phases of matter. The first of these projects is a treatment of an infinite dimensional generalization of the SSH model with local Coulomb interactions which is solved exactly using DMFT-NRG. Observed in the solution is power-law augmentation of the non-interacting density of states, as well as a Mott transition. This calculation represents an exact solution to an interacting topological insulator in the strongly correlated regime at zero temperature. The second set of projects involves the development of methods for formulating non-interacting auxiliary models for strongly correlated systems. These auxiliary models are able to capture the full dynamics of the original strongly correlated model, but with only completely non-interacting degrees of freedom, defined in an enlarged Hilbert space. We motivate the discussion by performing the mapping analytically for simple interacting systems using non-linear canonical transformations via a Majorana decomposition. For the nontrivial class of interacting quantum impurity models, the auxiliary mapping is established numerically exactly for finite-size systems using exact diagonalization, and for impurity models in the thermodynamic limit using NRG, both at zero and finite temperature. We find that the auxiliary systems take the form of generalized SSH models, which inherit the topological characteristics of those models. These generalized SSH models are also formalized and investigated in their own right as novel systems. Finally, we apply this methodology to study the Mott transition in the Hubbard model. In terms of the auxiliary system, we find that the Mott transition can be understood as a topological phase transition, which manifests as the formation and dissociation of topological domain walls.
... If you do not know anything regarding geometric algebra, I strongly recommend you [1]. You have a complete study of Geometric Algebra in [3]. ...
... See Fig.2 for a visual explanation. Also, in [1][2] and [3], you can find more information about the meaning or interpretation of the bivectors. If we multiply the three vectors, we obtain the trivector (also called pseudoscalar in the literature [1] [3]): ...
... Also, in [1][2] and [3], you can find more information about the meaning or interpretation of the bivectors. If we multiply the three vectors, we obtain the trivector (also called pseudoscalar in the literature [1] [3]): ...
In this paper, we will calculate the electromagnetic field strength and the Lorentz force in Geometric Algebra Cl(3,0). And we will compare it with their equivalent in the tensor covariant formalism. Leading to a one-to-one map in the four equations.
Also, we will obtain another four extra equations that do not appear in the classical formalism, and we will explain its meaning. In the same way, we will expand the Electromagnetic field strength elements and the velocity multivector of the particle.
This will lead to new equations and new elements appearing that will be explained. The most important one, is the Electromagnetic trivector Bxyz.
Lastly, an insight of the possible implications in the Dirac Equation of these acquirements will be commented.
... The very elegant expression (37) of the electromagnetic multivector F was used only by few authors like Arthur [20] and Macdonald [18] whereas most authors like Doran and Lasenby [21] or Hestenes [15] write the electromagnetic field multivector (35) in terms of the magnetic induction field pseudovector b. The choice made here seem much more natural in a sense since it reflects the underlying geometry of space. ...
... Taking into account the partial time derivative (38) and the gradient (40) of the electromagnetic multivector field and the definition (31) of the propagation velocity, we obtain, [21] 1 ...
... Using the constitutive relations (27) and (30) and the speed of light (31) the auxiliary electromagnetic multivector field G is recast as, [21] ...
We consider the electrodynamics of electric charges and currents in vacuum and then generalise our results to the description of a dielectric and magnetic material medium : first in spatial algebra (SA) and then in space-time algebra (STA). Introducing a polarisation multivector $\tilde{P} = \boldsymbol{\tilde{p}} -\,\frac{1}{c}\,\boldsymbol{\tilde{M}}$ and an auxiliary electromagnetic field multivector $G = \varepsilon_0\,F + \tilde{P}$, we express the Maxwell equation in the material medium in SA. Introducing a bound current vector $\tilde{J} = J -\,c\,\nabla\cdot\tilde{P}$ in space-time, the Maxwell equation is then expressed in STA. The wave equation in the material medium is obtained by taking the gradient of the Maxwell equation. For a uniform electromagnetic medium consisting of induced electric and magnetic dipoles, the stress-energy momentum vector is written as $\dot{T}\left(\dot{\nabla}\right) = \frac{1}{c}\,J \cdot F = f$ where $f$ is the electromagnetic force density vector in space-time. Finally, the Maxwell equation in the material medium can be written in STA as a wave equation for the potential vector $A$.
... which reduce to vacuum Maxwell's equations ∂ µ F µν = 0 in the case of electromagnetism. 6 Eq. (19) can also be derived by varying the action ...
... It is worth to note that expressing A µ in terms of V, through Eq. (3), the equations that result from variation of the action S Y M [V] are less restrictive (i.e., they allow more solutions) 6 The covariant derivative acts on matrix-valued fields (with gauge transformation M ′ = uMu † ) as ...
... Their shape-gauge image, Eq. (24), is a second-order equation for R (first-order for the shape operator S µ ), which is nonlinear also for the electromagnetic the- 8 We found particularly inspiring the ideas of Refs. [6,7,14], where the fundamental variable describing the geometry of a manifold is its pseudoscalar field, i.e., a multivector representation of the tangent space. ...
We introduce the concept of shape operator and rotating blade (also known in the theory of embedded Riemannian manifolds as the second fundamental form and the Gauss map) in the realm of Yang-Mills theories. Hence we arrive at new gauge-invariant variables, which can serve as an alternative to the usual gauge potentials.
... Space-time algebra [5,9] (isomorphic to Dirac algebra of quantum mechanics) is Clifford's geometric algebra Cl(3, 1) generated by all geometric products of R 3,1 vectors, also called (flat) Minkowski space, with a 16dimensional algebra basis of scalar, vector, bivector, trivector, and pseudoscalar elements: ...
... 3 Note that as for bivectors, the space-time duality map (2.6) exchanges relative vectors e t1 , e t2 , e t3 , with pure space bivectors e 12 , e 23 , e 31 , and vice versa. For Maxwell's theory [5,9] this means to exchange electric and magnetic fields. 4 Regarding physics, the split is determined by the time vector and its dual trivector (the three-dimensional space volume element). ...
... The special affine space-time Fourier transform (SASFT) maps 16-dimensional space-time algebra functions f ∈ L 1 (R 3,1 ; Cl(3, 1)) to 16-dimensional spectrum functions F{f } : R 3,1 → Cl (3,1). It is defined as 5 Note that Abe and Sheridan adopt in their 1994 papers that introduce the SAFT slightly different sign conventions in (61) of [1] and in (3) of [2]. For consistency, we use the conventions specified in (3) of [2]. ...
We generalize the space-time Fourier transform (SFT) (Hitzer in Adv Appl Clifford Algebras 17(3):497–517, 2007) to a special affine Fourier transform (SASFT, also known as offset linear canonical transform) for 16-dimensional space-time multivector Cl(3, 1)-valued signals over the domain of space-time (Minkowski space) \(\mathbb {R}^{3,1}.\) We establish how it can be computed in terms of the SFT, and introduce its properties of multivector coefficient linearity, shift and modulation, inversion, Rayleigh (Parseval) energy theorem, partial derivative identities, a directional uncertainty principle and its specialization to coordinates. All important results are proven in full detail.
... The construction of the universal real Clifford algebra is well-known, for details see e.g. [9,20]. We give only a brief description here. ...
... To make it clear we elaborate some examples for n = 2 in more detail. Example 3. 9 In the case of a 2-qubit we work in Clifford algebra C 4 of dimension is 2 4 = 16. Using the Witt basis f 1 , f † 1 , f 2 , f † 2 of C 4 we define a primitive idempotent I = f 1 f † 1 f 2 f † 2 which gives rise to spinor space S 2 = C 4 I of dimension 2 2 = 4 with an orthonormal basis ...
... We show two ways how to see a qubit in real Clifford algebra G 3 , i.e the GA induced by the standard euclidean inner product of signature (3, 0). The first approach appears in literature, see[6,9,19], and describes qubit states as even elements in this algebra or equivalently as unite quaternions. The second approach is ...
We propose to represent both n-qubits and quantum gates acting on them as elements in the complex Clifford algebra defined on a complex vector space of dimension 2n. In this framework, the Dirac formalism can be realized in straightforward way. We demonstrate its functionality by performing quantum computations with several well known examples of quantum gates. We also compare our approach with representations that use real geometric algebras.
... In this section, we present basis-free formulas for all characteristic polynomial coefficients in the geometric algebras G p,q , n = p + q = 5. The formula (5.8) for C (8) = −Det(U ) is presented in [25] and in some another form in [3]. The formula for C (1) = Tr(U ) is also presented in [25]. ...
... Let us consider the following well-known spin groups [8,13,19] Spin ...
... The geometric algebras of vector spaces of dimensions n = 4, 5, and 6 are important for different applications in physics (the space-time algebra G 1,3 [8,13,17], the conformal space-time algebras G 4,2 and G 2,4 [7,8]), in computer science and engineering (the conformal geometric algebra G 4,1 [4,5,10,14,18]), in computer vision and computer graphics (the geometric algebra G 3,3 of projective geometry [9,16]). In particular, the characteristic polynomial coefficients are used to solve the Sylvester and Lyapunov equations in geometric algebra [22,24]. ...
In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras \(\mathcal {G}_{p,q}\) of vector space of dimension \(n=p+q\). We present basis-free formulas for all characteristic polynomial coefficients in the cases \(n\le 6\), alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case \(n=4\), and one of the formulas in the case \(n=5\). We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade 1) and basis elements are presented in the case of arbitrary n, the formulas for rotors (elements of spin groups) are presented in the cases \(n\le 5\). The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation.
... Clifford algebras 1 [73,94,57] are regarded as a universal language for physics due to their intimate connection to geometry. They neatly geometrify algebra and algebraize geometry. ...
... The gamma matrices which are 4 × 4 anti-commuting unitary matrices solving the Dirac equation in quantum field theory gives a matrix representation for a Clifford algebra, Cl 1,3 (R) [116]. The Pauli I, X, Y, Z matrices which are quite significant for quantum mechanics and quantum computing provides a representation for the Clifford algebra Cl 0,3 (R) [57]. Schrödinger's equation can be represented as an element in the Clifford algebra Cl 0,1 (R) [80]. ...
This thesis develops the categorical proof theory for the non-compact multiplicative dagger linear logic, and investigates its applications to Categorical Quantum Mechanics (CQM). The existing frameworks of CQM are categorical proof theories of compact dagger linear logic, and are motivated by the interpretation of quantum systems in the category of finite dimensional Hilbert spaces. This thesis describes a new non-compact framework called Mixed Unitary Categories which can accommodate infinite dimensional systems, and develops models for the framework. To this end, it builds on linearly distributive categories, and $*$-autonomous categories which are categorical proof theories of (non-compact) multiplicative linear logic. The proof theory of non-compact dagger-linear logic is obtained from the basic setting of an LDC by adding a dagger functor satisfying appropriate coherences to give a dagger-LDC. From every (isomix) dagger-LDC one can extract a canonical "unitary core" which up to equivalence is the traditional CQM framework of dagger-monoidal categories. This leads to the framework of Mixed Unitary Categories (MUCs): every MUC contains a (compact) unitary core which is extended by a (non-compact) isomix dagger-LDC. Various models of MUCs based on Finiteness Spaces, Chu spaces, Hopf modules, etc., are developed in this thesis. This thesis also generalizes the key algebraic structures of CQM, such as observables, measurement, and complementarity, to MUC framework. Furthermore, using the MUC framework, this thesis establishes a connection between the complementary observables of quantum mechanics and the exponential modalities of linear logic.
... An interpretation of the power components in association with an equivalent circuit was presented. For the representation of power, the proposed model was based on the Geometric Algebra (GA) framework [16]. More specifically, a power multivector was introduced, which is fully capable of providing all the necessary information to determine the power components. ...
... By using bold notation to denote individual bivectorial terms, it can be written that (16) or, more concisely, ...
The identification of harmonic generating loads and the assignation of responsibility for harmonic pollution is an important first step for harmonic control in modern power systems. In this paper, a previously introduced power multivector is examined as a possible tool for the identification of such loads. This representation of power is based on the mathematical framework of Geometric Algebra (GA). Components of the power multivector derived at the point of connection of a load are grouped into a single quantity, which is a bivector in GA and is characterized by a magnitude, direction and sense. The magnitude of this bivector can serve as an indicator of the distortion at the terminals of the load. Furthermore, in contrast to indices based solely on magnitude, such as components derived from any apparent power equation, the proposed bivectorial representation can differentiate between loads that enhance distortion and those with a mitigating effect. Its conservative nature permits an association between the distortion at specific load terminals and the common point of connection. When several loads connected along a distribution line are considered, then an evaluation of the impact of each one of these loads on the distortion at a specific point is possible. Simulation results confirm that information included in the proposed bivector can provide helpful guidance when quantities derived from apparent power equations deliver ambiguous results.
... For instance, projective geometric algebra (PGA) G p,0,1 is useful for computations with flat objects and is applied in computer graphics and vision, robotics, motion capture, dynamics simulations [6,9,19,25,26]. PGA can be realized as a subalgebra of conformal geometric algebra (CGA) [18,29,32,33,35], which has applications in pose estimation, robotics, computer animation, machine learning, neural networks, etc. [20-22, 30, 31, 46]. The algebras G 3,0,1 , G 0,3,1 , even subalgebras G ...
... In the case of the complex geometric algebra G(C p+q,r ), the generators satisfy the same conditions but with the diagonal matrix η with p + q times 1 and r times 0 on the diagonal. Let us denote by Λ r := G 0,0,r the subalgebra of G p,q,r , which is the Grassmann (exterior) algebra [15,18,36]. Consider the subspaces G k p,q,r of grades k = 0, 1, . . . ...
In this paper, we introduce and study five families of Lie groups in degenerate Clifford geometric algebras. These Lie groups preserve the even and odd subspaces and some other subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups contain degenerate spin groups, Lipschitz groups, and Clifford groups as subgroups in the case of arbitrary dimension and signature. The considered Lie groups can be of interest for various applications in physics, engineering, and computer science.
... So far this is all well understood from the perspective of the "Lie Groups as Spin groups" paper 14 and chapter 11 of the Doran and Lasenby book. 15 However, we now want to give some specific details for SU (3), which go beyond the details presented in those sources, and then discuss the role of characteristic multivectors. We note that the approach being followed here as regards how the "Lie Groups as Spin groups" setup is implemented, differs somewhat from the Roelfs approach, in that here, as already described, we are taking the bivectors B as operating upon a concrete vector a in the 6d space. ...
... For electroweak, Hestenes, 17 Antony Lewis (in unpublished notes) and Chris Doran and myself 15 have pointed out that given an STA Dirac spinor (which for electroweak will generally be some combination of massless left-handed electron and neutrino wavefunctions); then starting from a Lagrangian, we find that the symmetry we should look for is to find all multivectors N such that when we carry out the transformation → e N , then 0̃, the Dirac current, is invariant. This picks out the set of bivectors, which commute with 0 , i.e., I 1 , I 2 and I 3 , and the pseudoscalar I, which reverses to itself, but anticommutes with 0 . ...
Different ways of representing the group SU(3)$$ SU(3) $$ within a Geometric Algebra approach are explored. As part of this, we consider characteristic multivectors for SU(3)$$ SU(3) $$ and how these are linked with decomposition of generators into commuting bivectors. The setting for this work is within a 6d Euclidean Clifford Algebra. We then go on to consider whether the fundamental forces of particle physics might arise from symmetry considerations in just the 4d geometric algebra of spacetime—the STA. As part of this, a representation of SU(3)$$ SU(3) $$ is found wholly within the STA, involving preservation of a bivector norm. We also show how Octonions can be fully represented within the Spacetime Algebra, which we believe will be useful in making them understandable and accessible to a new community in Physics and Engineering. The two strands of the paper are drawn together in showing how preserving the octonion norm is the same as preserving the timelike part of the Dirac current of a particle. This suggests a new model for the symmetries preserved in particle physics. Following on from work by Günaydin and Gürsey on the link between quarks, and octonions, and by Furey on chains of octonionic multiplications, we show how both of these fit well within our scheme and give some wholly STA versions of the operations involved, which in the cases considered have easily understandable equivalents in terms of 4d geometry. Links with larger groups containing SU(3)$$ SU(3) $$, such as G2$$ {G}_2 $$ and SU(8)$$ SU(8) $$, are also considered.
... Relations (4,5) define the spacetime algebra, STA of Hestenes [7,8] -a real 16D Clifford algebra ℓ (1,3) ((1,3) stands for the signature, see (3)) generated by the action of the geometric product onto the 4-vectors of the real 4D spacetime. Hestenes also proposed an STA DE [6,7] without matrices and with the complex structure arising from the (multi)vectors of STA alone. ...
... We can lift indices up and down, i.e. swap between reciprocal bases in X, by the appropriate form of the signature in (8). The quantization postulate STR yields the STR DE ( -rest mass of the electron): ...
Dirac's leaping insight that the normalized anti-commutator of the {\gamma}^{\mu} matrices have to equal the relativistic timespace signature was decisive for the successful formulation of his famous Equation. The Dirac matrices represent 'some internal degrees of freedom of the electron' and are the same in all Lorentz frames. Therefore, the link to the timespace signature of special relativity constitutes a separate postulate of Dirac's theory. I prove in this contribution that all the properties of the Dirac electron & positron follow from the quantization of the relativistic 4-momentum vector - preconceived 'internal degrees of freedom', matrices and imposed signature unneeded. The proposed formalism is powerful and clearly rooted in physical spacetime augmented with handedness / reflection.
... Here, I address spin in the nonrelativistic regime under the spin-position decoupling approximation. 6,7,[11][12][13] The 3D physical space in this case reduces to orientation space at a point (the origin), and the relevant symmetry operations are proper rotations and reflections. In STR, it corresponds to the subspace Σ in (5) 12 with a Clifford algebra Cl 3, 0 ð Þ isomorphic to that of Pauli matrices, therefore the notation. ...
... Within a normalization factor, the two first expressions of spin in (12) are identical to the definition of spin vector in STA. 7 The corresponding expression for the full spin model (8) is the first equation in (10). It is also clear from Figure 1A,B that any unit spin vector u can be expressed by means of the two unit basis spins AEσ 3 through the "tailormade" linear combination of the orthonormal "spinor" pair u þ ,u À . ...
Under the spin–position decoupling approximation, a vector with a phase in 3D orientation space endowed with geometric algebra substitutes the vector–matrix spin model built on the Pauli spin operator. The standard quantum operator‐state spin formalism is replaced with vectors transforming by proper and improper rotations in the same 3D space—isomorphic to the space of Pauli matrices. In the single‐spin case, the novel spin 1/2 representation (1) is Hermitian, (2) shows handedness, (3) yields all the standard results and its modulus equals the total spin angular momentum Stot=3ℏ/2$$ {S}_{tot}=\sqrt{3}\hslash /2 $$, (4) formalizes irreversibility in measurement, and (5) permits adaptive imbedding of the 2D spin space in 3D. Maximally entangled spin pairs (1) are in phase and have opposite handedness, (2) relate by one of the four basic improper rotations in 3D: plane reflections (triplets) and inversion (singlet), (3) yield the standard total angular momentum, and (4) all standard expectation values for bipartite and partial observations follow. Depending on whether proper and improper rotors act one—or two—sided, the formalism appears in two complementary forms, the “spinor” or the “vector” form, respectively. The proposed scheme provides a clear geometric picture of spin correlations and transformations entirely in the 3D physical orientation space.
... For the moment, let us suppose this is not the case. Now suppose that our theorem (that P commutes and anticommutes with an equal number of blades) is true for n − 1; P must therefore commute with N∕4 elements of A and anticommute with N∕4 elements of A. Call these subsets A (1) and A (2) respectively. Since P does not contain e , then elements of the form e A (1) will commute with P if P is even, and anticommute with P if P is odd. ...
... Since P does not contain e , then elements of the form e A (1) will commute with P if P is even, and anticommute with P if P is odd. Similarly, elements of the form e A (2) will anticommute with P if P is even, and commute with P if P is odd. Thus, for P either even or odd, we see that the subset B is divided equally into elements that commute and anticommute with P. If the theorem is true for n − 1 it is therefore true for n as well. ...
If an initial frame of vectors {ei}$$ \left\{{e}_i\right\} $$ is related to a final frame of vectors {fi}$$ \left\{{f}_i\right\} $$ by, in geometric algebra (GA) terms, a rotor, or in linear algebra terms, an orthogonal transformation, we often want to find this rotor given the initial and final sets of vectors. One very common example is finding a rotor or orthogonal matrix representing rotation, given knowledge of initial and transformed points.
In this paper, we discuss methods in the literature for recovering such rotors and then outline a GA method, which generalises to cases of any signature and any dimension, and which is not restricted to orthonormal sets of vectors. The proof of this technique is both concise and elegant and uses the concept of characteristic multivectors as discussed in the book by Hestenes and Sobczyk, which contains a treatment of linear algebra using geometric algebra. Expressing orthogonal transformations as rotors, enables us to create fractional transformations and we discuss this for some classic transforms. In real applications, our initial and/or final sets of vectors will be noisy. We show how to use the characteristic multivector method to find a ‘best fit’ rotor between these sets and compare our results with other methods.
... If you do not know what I am talking about, I strongly recommend you check the masterpiece [1] and the best collection of Geometric Algebra knowledge [3]. ...
... The Maxwell Equations in Geometric Algebra Cl3.0We want to reproduce the same results as in chapter 7 (34) and (36) but using Geometric Algebra Cl3,0. In[3] (7.14) the Maxwell Equation is reduced to this form Now, we can apply the same equation using all the extended definition we have commented for each one of the elements(20)(26)(27) in (37).I represent in bold, the elements that do not exist in standard algebra (nor in covariant tensor formulation neither in standard Gibbs-Heaviside vector calculations): ...
In this paper, we will obtain the Maxwell’s equations using Geometric Algebra Cl3,0 (three space dimensions and the time as the trivector of the Algebra). We will obtain the equations with all the possible elements that exist in this Algebra.
We will get as new elements the Electromagnetic Bivector Bxyz and the ones related to the angular momentum of the charge (in relation to its rotation or helicoidal trajectory).
Considering these new elements zero or oscillatory with a zero average, we get the standard Maxwell’s Equations (both in its covariant formulism and in standard vector algebra.)
... In view of relations (5) and (6), the vector v is resolved into parallel and perpendicular parts (Fig. 1), ...
... We consider an orthonormal vector spatial frame {ê 1 ,ê 2 ,ê 3 }. The geometric product of two basis vectors reads, [5] e iêj =ê i ·ê j +ê i ∧ê j with i, j = 1, 2, 3 (A.1) ...
In geometric algebra, the rotation of a vector is described using rotors. Rotors are phasors where the imaginary number has been replaced by a oriented plane element of unit area called a unit bivector. The algebra in three dimensional space relating vectors and bivectors is the Pauli algebra. Multivectors consisting of linear combinations of scalars and bivectors are isomorphic to quaternions. The rotational dynamics can be expressed entirely in the plane of rotation using bivectors. In particular, the Poisson formula providing the time derivative of the unit vectors of a moving frame are recast in terms of the angular velocity bivector and applied to cylindrical and spherical frames. The rotational dynamics of a point particle and a rigid body are fully determined by the time evolution of rotors. The mapping of the angular velocity bivector onto the angular momentum bivector is the inertia map. In the principal axis frame of the rigid body, the inertia map is characterised by symmetric coefficients representing the moments of inertia. The Huygens-Steiner theorem, the kinetic energy of a rigid body and the Euler equations are expressed in terms of bivector components. This formalism is applied to study the rotational dynamics of a gyroscope.
... In the following it will be shown that actual weirdness of all conventional quantum mechanics comes from logical inconsistence of what is meant in basic quantum mechanical definitions and has nothing to do with the phenomena scale and the attached artificial complementarity principle [1] [2] [3] [4]. ...
... A major theorem, first proved by Brauer and Weyl (1935) [3], is that the algebra of outer products of spinors is isomorphic to the Clifford algebra [6,11,12] of multivectors, (see [17] for a history), also known as the geometric algebra [9,13,15,16], in any number of spacetime dimensions. ...
Spinors are central to physics: all matter (fermions) is made of spinors, and all forces arise from symmetries of spinors. It is common to consider the geometric (Clifford) algebra as the fundamental edifice from which spinors emerge. This paper advocates the alternative view that spinors are more fundamental than the geometric algebra. The algebra consisting of linear combinations of scalars, column spinors, row spinors, multivectors, and their various products, can be termed the supergeometric algebra. The inner product of a row spinor with a column spinor yields a scalar, while the outer product of a column spinor with a row spinor yields a multivector, in accordance with the Brauer–Weyl (Am J Math 57: 425–449, 1935, https://doi.org/10.2307/2371218) theorem. Prohibiting the product of a row spinor with a row spinor, or a column spinor with a column spinor, reproduces the exclusion principle. The fact that the index of a spinor is a bitcode is highlighted.
... Motivation. The quaternions H is an interesting object not only in pure mathematics (e.g., [5], [10], [11], [12], [13] [14], [17], [19], [23]), but also in applied mathematics (e.g., [4], [7], [15], [16], [20] and [21]). Independently, spectral analysis on H is considered in [2] and [3], under representation, "over C," different from the usual quaternion-eigenvalue problems of quaternion-matrices studied in [13], [15] and 16[]. ...
In this paper, we consider natural Hilbert-space representations $\left\{ \left(\mathbb{C}^{2},\pi_{t}\right)\right\} _{t\in\mathbb{R}}$ of the hypercomplex system $\left\{ \mathbb{H}_{t}\right\} _{t\in\mathbb{R}}$, and study the realizations $\pi_{t}\left(h\right)$ of hypercomplex numbers $h\in\mathbb{H}_{t}$, as $\left(2\times2\right)$-matrices acting on $\mathbb{C}^{2}$, for an arbitrarily fixed scale $t\in\mathbb{R}$. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.
... Regardless of how the pairwise amino acid orientation is captured (interatomic distances, angle maps, rigid body frames, etc) it has been demonstrated that it represents key information when trying to resolve the overall 3D protein structure. We believe that a suitable candidate for providing information on pairwise orientations of amino acid is represented by Geometric Algebra (GA), due to its intuitive handling of geometrical objects and operations on them 23,24 . GA has already found some applications in protein modelling: for example non-normalized rotors (see Section 3 for further details) have been employed for tackling the loop closure problem, i.e. when feasible loops must be found between two given anchor residues, and to compute atomic coordinates from internal coordinates 25,26 . ...
Extended version of the paper presented at the ENGAGE 2022 conference. We introduce two novel GA-based metrics which contain information on relative orientations of amino acid residues. We then employ these metrics as an additional input features to a Graph Transformer (GT) architecture to aid the prediction of the 3D coordinates of a protein, and compare them to classical angle-based metrics. We show how our GA features yield comparable results to angle maps in terms of accuracy of the predicted coordinates. This is despite being constructed from less initial information about the protein backbone. The features are also fewer and more informative, and can be (i) closely associated to protein secondary structures and (ii) more readily predicted compared to angle maps. We hence deduce that GA can be employed as a tool to simplify the modeling of protein structures and pack orientational information in a more natural and meaningful way.
... We start from the second principle of mechanics, written with the help of the momentum ⃗ F = d⃗ p dt (1) and we express the kinetic energy W dW = ⃗ F d⃗ r (2) according to the kinetic energy variation theorem. Further, we obtain: ...
We start from the second principle of mechanics, written with the help of the momentum ⃗ F = d⃗ p dt (1) and we express the kinetic energy W dW = ⃗ F d⃗ r (2) according to the kinetic energy variation theorem. Further, we obtain: We denote everywhere in this article, the scalar product with point or juxtaposition, and the square of a vector will be the scalar product of the vector with itself −dW dt + d⃗ p d⃗ r = 0 (3) But, we have established in relativistic kinematics the expression cdt⃗ e ⊕ d⃗ r = d⃗ x (4) introducing the notions of spacetime, quadravector and Lorentz transformation. c is the speed of light in vacuum, and ⃗ e is the unit vector on the Ot axis, with property ⃗ e 2 = −1. The operator ⊕ is the direct sum between vector spaces. Since 0 is a scalar and d⃗ x is a quadravector, the following expression is also a quadravector. d ⃗ P = dW c ⃗ e ⊕ d⃗ p (5) and by integration... ⃗ P = W c ⃗ e ⊕ ⃗ p (6) since relation (3) can be written d ⃗ P · d⃗ x = dW c ⃗ e ⊕ d⃗ p (cdt⃗ e ⊕ d⃗ r) = −dW dt + d⃗ p · d⃗ r = 0 1
... Furthermore, geometric algebras enable powerful geometric predicates and modules [18], providing, if used with caution, performance which is on par with the current state-ofthe-art frameworks [27]. In the past decades, these algebras have proven to be able to solve a variety of problems in various fields, involving inverse kinematics [13] and physics [5]. In conclusion, this work constitutes yet another step towards an effective all-in-one geometric algebra framework for handling VR data streams, with performance that is on par or exceeds current SoA frameworks. ...
As shared, collaborative, networked, virtual environments become increasingly popular, various challenges arise regarding the efficient transmission of model and scene transformation data over the network. As user immersion and real-time interactions heavily depend on VR stream synchronization, transmitting the entire data set does not seem a suitable approach, especially for sessions involving a large number of users. Session recording is another momentum-gaining feature of VR applications that also faces the same challenge. The selection of a suitable data format can reduce the occupied volume, while it may also allow effective replication of the VR session and optimized post-processing for analytics and deep-learning algorithms. In this work, we propose two algorithms that can be applied in the context of a networked multiplayer VR session, to efficiently transmit the displacement and orientation data from the users’ hand-based VR HMDs. Moreover, we present a novel method for effective VR recording of the data exchanged in such a session. Our algorithms, based on the use of dual-quaternions and multivectors, impact the network consumption rate and are highly effective in scenarios involving multiple users. By sending less data over the network and interpolating the in-between frames locally, we manage to obtain better visual results than current state-of-the-art methods. Lastly, we prove that, for recording purposes, storing less data and interpolating them on-demand yields a data set quantitatively close to the original one.
... Let us note that there is also notation n = e − + e + andn = e + − e − instead of e 0 , e ∞ , respectively. 9 Generally in geometric algebra models, we represent objects by so-called OPNS and IPNS representations with the help of dot and wedge product. In the detail, for each point P and each geometric object O in IPNS representation (the geometric object O * in OPNS representation, respectively), the following property holds ...
We will study binocular vision for a 6‐DOF robotic manipulator in the conformal geometric algebra. We will focus on a case where some pieces of information, such as relative cameras positions, have been lost. In particular, we will use the construction of the manipulator to infer a self‐calibration method for cameras position based on binocular vision with incomplete information.
... Pseudo-Hermitian systems have been studied in different contexts ranging from the systems with complex potentials, resonance phenomena associated to nuclear, atomic or molecular systems, nano-structured materials or condensates, to even the systems which are not so quantum mechanical in sense but their physical behaviour is quite amenable to quantum language (for example classical statistical mechanical systems, biological systems with diffusion, light propagation in wave guides) and many other fields where even the conventional quantum mechanics has already shown success [10,11,12]. On the other hand, apart from the development of formal methods [1,13] bi-orthogonal system has got special attention in the study of super-symmetry, Lie super-algebra and quantum mechanics over Galois field etc. [14,15,16,17] The present report deals with a non-Hermitian version of so called Pauli Hamiltonian that describes the behaviour of a single electron in a magnetic field B. The usual expression of interaction Hamiltonian H int " σ¨ B [18,19] has a 2ˆ2 matrix representation which is Hermitian in the conventional sense of innerproduct xu|vy " u ‹ 1 v 1`u particular and geometric algebra in general have been widely used in formulating various physical theories [23] like classical mechanics [24], electrodynamics [25], relativity [26] and even in current areas of interest like quantum information theory [27]. Unlike its Hermitian counterpart our present Hamiltonian is written in terms of a new set of generators obtained from a set of bi-orthogonal vectors and two of them are non-Hermitian in conventional sense. ...
A set of two-parameter bi-orthogonal eigen-spinors has been constructed from a deformed pseudo- Hermitian extension of Pauli Hamiltonian and its Hermitian conjugate. The Hamiltonians thus obtained are iso-spectral to the original Pauli Hamiltonian. A pair of spin-projection operators has been constructed as an essential ingredient of a possible bi-orthogonal quantum mechanics. An analogue of Kramers theorem in pseudo-Hermitian setting has also been inferred in a conjectural sense. The properties of time reversal and bi-orthogonality have been elaborated in the frame work of Clifford algebra Cl3, where the spinors have been viewed as elements of left ideal and the relevant inner-products are understood in terms of different involutions leading to elements of a division ring. The whole process of present construction is found to be based on both direct and time reversed Cl3 generators. A new variant of Kustaanheimo-Stiefel transformation has been introduced with the help of spinor operator.
... If you do not know what I am talking about, I strongly recommend you check the masterpiece [1] and the best collection of Geometric Algebra knowledge [3]. ...
In this paper, we will calculate a generalization of the Dirac Equation using Geometric Algebra Cl3,0. Apart from the partial derivatives with respect to position and time, also partial derivatives regarding orientation (or angular momentum) will appear.
The reason that these new partial derivatives have not been considered before is probably because their value is very small or directly zero or because they represent an oscillatory movement or value which mean value is zero. Meaning they can influence in local effects (helicoidal movement, rotations etc.) but not in the mean trajectory of the particles.
... GA is a suitable candidate to represent the features mentioned above due to its intuitive handling of geometrical objects and operations on them [15,16]. GA has already found some applications in protein modelling, especially in the molecular distance problem [17,18], but to the best of our knowledge there has not been an effort to employ GA modelling for PSP. ...
0000−0001−8775−4427] , Joan Lasenby 1[0000−0002−0571−0218] , and Pablo Chacón 2[0000−0002−3168−4826] Abstract. The state of the art in protein structure prediction (PSP) is currently achieved by complex deep learning pipelines that require several input features. In this paper, we demonstrate the relevance of Geometric Algebra (GA) for modelling protein features in PSP. We do so by proposing a novel GA metric based on the relative orientations of amino acid residues. We then employ this metric as an additional input feature to a Graph Transformer (GT) to aid the prediction of the 3D coordinates of a protein. Adding this GA-based orientational information improves the accuracy of the predicted coordinates even after few learning iterations and on a small dataset.
... A detailed study can be seen in the work of Goldenfeld et al.22 Renormalization group techniques allow the study of a physical system at different scales, so that they are intimately related with scaling symmetry, DA, and self-similarity.Geometric algebra (GA) is a very general approach to the formulation of mathematical physics also related to DA. An excellent source to learn GA is the work of Doran et al.23 It represents one of the most ambitious unification approaches to the concepts of Vector Analysis, Algebra, and Geometry, allowing an intermediate perspective between purely synthetic (axiomatic geometry) and purely analytic (coordinate based geometry) formulations trying to preserve the advantages of both approaches.Sometimes, a clever combination of the election of variables of a problem, its simplification hypotheses, and DA is enough to obtain the complete mathematical dependence of the solution of the problem, except some undefined constants of the order of unity in most cases.The aim of physical theories is, in fact, to refine the value of these constants and to provide a conceptual framework to understand the mathematical formulation.The most important result in DA is the Buckingham Pi theorem, which says 11 : every physical equation with a certain number n of physical variables, can be rewritten in terms of a set of p = n − k dimensionless parameters π 1 , π 2 , . . . ...
The model based systems engineering (MBSE) approach describes a system using consistent views to provide a holistic model as complete as possible. MBSE methodologies end with the physical architecture of the system, but a physical model is clearly incomplete without the study of its associated physical laws and phenomena related to the whole system or its parts. However, the computational demands could be excessive even for modest projects. Dimensional analysis (DA) is common in fluid dynamics and chemical engineering, but its application to systems engineering is still limited. We describe an engineering methodological process, which incorporates DA as a powerful tool to understand the physical constraints of the system without the burden of complex analytical or numerical calculations. A detailed example describing a microantenna is presented showing the benefits of this approach. The selected example describes a problem rarely covered in modern expositions of DA in order to show the wide benefit of these techniques. The information provided by this analysis is very useful to select the best physically realizable architectures, testing design, and conduct trade-off studies. The complexity of modern systems and systems of systems demands new testing procedures in order to comply with increasingly demanding requirements and regulations. This can be accomplished through research in new DA methods. Finally, this article serves as a fairly comprehensive guide to the use of DA in the context of MBSE, detailing its strengths, limitations, and controversial issues
... In this paper, we will focus on representing rotations in the Euclidean space in terms of rotors (4D representation) and bivectors (3D representation), and on two kinds of rotor-to-bivector maps, namely, (i) the exponential map, for which B = −2 log R and (ii) the Cayley transform, for which B = (1 − R)∕(1 + R). For a more detailed analysis, we refer the reader to Chapters 2 and 4 of Doran et al. 24 ...
Many problems in computer vision today are solved via deep learning. Tasks like pose estimation from images, pose estimation from point clouds or structure from motion can all be formulated as a regression on rotations. However, there is no unique way of parametrizing rotations mathematically: matrices, quaternions, axis‐angle representation or Euler angles are all commonly used in the field. Some of them, however, present intrinsic limitations, including discontinuities, gimbal lock or antipodal symmetry. These limitations may make the learning of rotations via neural networks a challenging problem, potentially introducing large errors. Following recent literature, we propose three case studies: a sanity check, a pose estimation from 3D point clouds and an inverse kinematic problem. We do so by employing a full geometric algebra (GA) description of rotations. We compare the GA formulation with a 6D continuous representation previously presented in the literature in terms of regression error and reconstruction accuracy. We empirically demonstrate that parametrizing rotations as bivectors outperforms the 6D representation. The GA approach overcomes the continuity issue of representations as the 6D representation does, but it also needs fewer parameters to be learned and offers an enhanced robustness to noise. GA hence provides a broader framework for describing rotations in a simple and compact way that is suitable for regression tasks via deep learning, showing high regression accuracy and good generalizability in realistic high‐noise scenarios.
... with corresponding elements of S and S † using identifications (4) and (5). Thus, any n-gate lies in the tensor product of a spinor space and its dual space S ⊗ S † . ...
This paper introduces an application of complex Clifford algebra in a representation of the quantum prisoner's dilemma. The authors propose a novel modification of the Eisert–Lewenstein–Wilkens protocol to represent a repeated version of the quantum game. This repeated modification allows to embed entanglement into players' strategy sets and to see how players will operate with it. The apparatus of complex Clifford algebra enables an intuitive representation of the suggested protocol and efficient computation of the resulting payoff functions. The presented findings provide a new point of view on the interpretation of entanglement as a measure of information transition between rounds of the game.
... The unification of vector, scalar, dimension, and geometric operations makes it possible to apply geometric algebra for the expression of geographic objects and compute the spatial relations. The research on the application of geometric algebra has matured around the world [27][28][29][30], and there are various research results in the theoretical physics [31,32], mathematics [33,34], computer science [35], and geoscience [36][37][38][39][40][41][42][43][44][45][46][47][48] fields. ...
The study of spatial geometric similarity plays a significant role in spatial data retrieval. Many researchers have examined spatial geometric similarity, which is useful for spatial analysis and data retrieval. However, the majority of them focused on objects of the same type. Methods to support the spatial geometric similarity computation for different types of objects are rare, a systematic theory index has not been developed yet, and there has not been a comprehensive computational model of spatial geometric similarity. In this study, we conducted an analysis of the spatial geometric similarity computation based on conformal geometric algebra (CGA), which has certain advantages in the quantitative computation of the measurement information of spatial objects and the qualitative judgment of the topological relations of spatial objects. First, we developed a unified expression model for spatial geometric scenes, integrating shapes of objects and spatial relations between them. Then, we established a model for the spatial geometric similarity computation under various geographical circumstances to provide a novel approach for spatial geometric similarity research. Finally, the computation model was verified through a case study. The study of spatial geometric similarity sheds light on spatial data retrieval, which has scientific significance and practical value.
In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta formulas with the ordinary Vieta formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand–Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis‐free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering.
A gas in a box is perhaps the most important model system studied in thermodynamics and statistical mechanics. Usually, studies focus on the gas, whereas the box merely serves as an idealized confinement. The present article focuses on the box as the central object and develops a thermodynamic theory by treating the geometric degrees of freedom of the box as the degrees of freedom of a thermodynamic system. Applying standard mathematical methods to the thermodynamics of an empty box allows equations with the same structure as those of cosmology and classical and quantum mechanics to be derived. The simple model system of an empty box is shown to have interesting connections to classical mechanics, special relativity, and quantum field theory.
A statistical simulation is presented which reproduces the correlation obtained from EPR coincidence experiments without non-local connectivity. Defining spin under the quaternion group reveals hyper-helicity, a hitherto missed attribute of spin. Including this in the treatment, reveals two complementary properties: spin polarization and spin coherence. The former has a CHSH value of 2, and spin coherence has a CHSH = 1 giving CHSH = 3 for an EPR pair. The simulation here gives 2.995. We suggest that Nature has CHSH=3 being considerable more than predicted from quantum mechanics of $2\sqrt{2}$. There are no Local Hidden Variables. We suggest that quaternion spin is more fundamental than Dirac spin. A computer program which performs the simulation without non-local connectivity is described.
We present an analysis of the local singlet state and compare its correlation to that of a separated pair of spins devoid of entanglement. In addition to polarization, we include its hyper-helicity, and this quantum coherence is shown to account for the quantum correlation that leads to the apparent violation of Bell's inequalities. Upon separation, particles conserve linear momentum, angular momentum, helicity and correlation. In order to agree with experimental coincidence data, both polarization and coherence, incompatible elements of reality, must simultaneously exist.
We explore the \({\mathbb Z}_2\) graded product \(C{\ell }_{10} = C{\ell }_4 \, {\widehat{\otimes }} \, C{\ell }_6\) as a finite internal space algebra of the Standard Model of particle physics. The gamma matrices generating \(C{\ell }_{10}\) are expressed in terms of left multiplication by the imaginary octonion units and the Pauli matrices. The subgroup of Spin(10) that fixes an imaginary unit (and thus allows to write \({\mathbb O} = {\mathbb C} \oplus {\mathbb C}^3\) expressing the quark-lepton splitting) is the Pati-Salam group \(G_\textrm{PS} = Spin (4) \times Spin (6) / {\mathbb Z}_2 \subset Spin (10)\). If we identify the preserved imaginary unit with the \(C{\ell }_6\) pseudoscalar \(\omega _6 = \gamma _1 \cdots \gamma _6\), \(\omega _6^2 = -1\), then \(\mathcal{P} = \frac{1}{2} (1 - i\omega _6)\) will be the projector on the extended particle subspace, including the right-handed (sterile) neutrino. We express the generators of \(C{\ell }_4\) and \(C{\ell }_6\) in terms of fermionic oscillators \(a_{\alpha } , a_{\alpha }^* , \alpha = 1,2\) and \(b_j , b_j^* , j = 1,2,3\) describing flavour and colour, respectively. The internal space observables belong to the Jordan subalgebra of hermitian elements of the complexified Clifford algebra \({\mathbb C} \otimes C{\ell }_{10}\) which commute with the weak hypercharge \(\frac{1}{2} Y = \frac{1}{3} \sum _{j=1}^3 b_j^* b_j - \frac{1}{2} \sum _{\alpha = 1}^2 a_{\alpha }^* a_{\alpha }\). We only distinguish particles from antiparticles if they have different eigenvalues of Y. Thus the sterile neutrino and antineutrino (both with \(Y=0\)) are allowed to mix into Majorana neutrinos. Restricting \(C{\ell }_{10}\) to the particle subspace, which consists of leptons with \(Y < 0\) and quarks, allows a natural definition of the Higgs field \(\varPhi \), the scalar of Quillen’s superconnection, as an element of \(C{\ell }_4^1\), the odd part of the first factor in \(C{\ell }_{10}\). As an application we express the ratio \(\frac{m_H}{m_W}\) of the Higgs and the W-boson masses in terms of the cosine of the theoretical Weinberg angle.
This study proposes an explanation for the physical power flow in planar circuits by analogy to theoretical two-dimensional circuits using a new mathematical model based on Geometric Algebra (GA) and 2D Maxwell’s equations. In contrast with traditional 3D physics in the observable real world, the magnetic field can be defined as a bivector instead of an axial vector allowing to obtain the Poynting Vector directly in a 2D flat world, where physical variables of planar circuits can be obtained. This approach is presented here for the first time to the best of the author’s knowledge. Previous investigations have focused on simplifications and symmetries of real 3D circuits studied mainly in the phasor and frequency domain. In this work, the electromagnetic power flow phenomenon is analyzed on a completely 2D time-domain basis and derived directly from the undisputed Maxwell equations, formulated in two dimensions. Several cases of special interest in AC multi-phase circuits are presented using the proposed technique, bringing a new simplified approach to the measurement of power flow exchange between the source and the load. It suggests a new way to understand energy propagation from a purely physical point of view.
A rigid body motion, which can be decomposed into rotation and translation, is essential for engineers and scientists who deal with moving systems in a space. While translation is as simple as vector addition, rotation is hard to understand because rotations are non-Euclidean, and there are many ways to represent them. Additionally, each representation comes with complex operations, and the conversions between different representations are not unique. Therefore, in this tutorial we review rotation representations which are widely used in industry and academia such as rotation matrices, Euler angles, rotation axis-angles, unit complex numbers, and unit quaternions. In particular, for better understanding we begin with rotations in a two dimensional space and extend them to a three dimensional space. In that context, we learn how to represent rotations in a two dimensional space with rotation angles and unit complex numbers, and extend them respectively to Euler angles and unit quaternions for rotations in a three dimensional space. The definitions and properties of mathematical entities used for representing rotations as well as the conversions between various rotation representations are summarized in tables for the reader’s later convenience.
Geometric algebra (GA) is proposed as a mathematical framework for revisiting fundamental aspects of sliding mode control (SMC) in nonlinear, switch‐controlled, single input systems. Sliding mode existence conditions, the switching policy, the invariance conditions, the associated equivalent control, and the characterization of ideal sliding dynamics, are all re‐examined using a geometric algebra (GA) standpoint. Two illustrative examples, from switched power electronics, are presented using the GA language.
A non-orthogonal frame is proposed in this work as an alternative representation of moving frames, which can describe naturally and conveniently orientations of the mobile platform of parallel pointing mechanisms. Fundamentals of non-orthogonal bases are presented, upon which a non-orthogonal frame is established utilizing two body-attached vectors and their bivector. Properties of the non-orthogonal frame are analyzed geometrically and algebraically. With the alternative representation, we revisit the kinematics of parallel pointing mechanisms. Efficient and robust kinematic formulations are obtained and demonstrated with examples.
This article presents a short review of a geometric approach to the theory of elementary particles which was proposed early. The wave function of particle, instead of the spinor representation, is given by the Clifford number, whose transfer rules have the structure of the Dirac equation for any manifold. The solution to this equation is obtained in terms of geometric characteristics. New experiments are proposed to show the geometric nature of the wave function of an elementary particle.
Provide a proof to the sufficiency of the existence of an invariant speed, that is identical experimentally to the speed of light in vacuum, so that it can be measured.
The expeditious market transformation to smart portable devices has created an opportunity to support activity recognition using the embedded sensors of these devices. Over the last decade, many activity recognition approaches have been proposed for various activities in different settings. The motion mode recognition or transition in modes of the device is needed in many technological domains. This approach detects a variety of motion modes for a human using a portable device. The approach includes many aspects: usability, mounting and data acquisition, sensors used, signal processing, methods employed, features extracted, and classification techniques. This chapter sums up with a comparison of the performance of several motion mode recognition techniques. In this research, multiple behaviors were distinguished using embedded inertial sensors in portable smart devices. In our experiments, we selected four types of human activity, which are walking, standing, sitting, and running. A combination of one of the embedded mobile sensors and machine learning techniques have been proposed in order to do this kind of classification. The proposed system relies on accelerometer data to classify user activities. The results show that using SVM classifier showed better accuracy for detection compared to the outcomes of the other classifiers like KNN and ensemble classifiers. For future work, classification of other human activities like cycling, driving, and swimming will be investigated
While the Maxwell’s equations describe the electric and magnetic fields developed by electrically charged particles and/or currents, the Lorentz force law completes the picture of classical electromagnetic theory by defining the force acting on a localised charge (or charge distribution) moving in the field. However, the standard formulation using vector algebra suffers from several inadequacies and unwarranted features. Clifford’s geometric algebra or more specifically space–time algebra, i.e geometric algebra in 4D Minkowski space–time, provides an elegant, compactified and comprehensive description by removing the discrepancies of the earlier formulation. It provides an invariant description, in the appropriate space–time setting, in terms of the combined electromagnetic field without reference to any inertial system. Moreover, using elementary geometric calculus, it facilitates direct analytical introduction of the putative concept of magnetic monopole and renders the equations for both the constituent fields, symmetric and inhomogeneous. In terms of the single space–time force equation, space–time algebra also encapsulates both the Lorentz force equation and the electromagnetic power equation.
Spin group and screw algebra, as extensions of quaternions and vector algebra, respectively, have important applications in geometry, physics and engineering. In three-dimensional projective geometry, when acting on lines, each projective transformation can be decomposed into at most three harmonic projective reflections with respect to projective lines, or equivalently, each projective spinor can be decomposed into at most three orthogonal Minkowski bispinors, each inducing a harmonic projective line reflection. In this paper, we establish the corresponding result for three-dimensional affine geometry: with each affine transformation is found a minimal decomposition into general affine reflections, where the number of general affine reflections is at most three; equivalently, each affine spinor can be decomposed into at most three affine Minkowski bispinors, each inducing a general affine line reflection.
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.
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