Quaternion algebra

Equation (9) shows that real numbers, the algebra of complex numbers, and the algebra of quaternions. (A proof of this statement may be found in Littlewood (1958), p. 251.)... 

Gibbs also wrote an influential book on statistical mechanics, which developed a molecular theory of thermodynamic properties from first principles, with a treatment that was general enough to accommodate quantum mechanics, a theory that had not even been imagined yet. He championed the now standard use of vector notation over William Rowan Hamilton s quaternion algebra and wrote several seminal papers on electromagnetism in the 1880s that supported Maxwell s theory, see also Equilibrium Maxwell, James Clerk Thermodynamics. 

In this section we will revisit KS regularization and present a new, elegant derivation of it, using quaternion algebra and an unconventional conjugate u referred to as the star conjugate of the quaternion u = uq + iu +j U2 + ku3 ... 

In the preceding text we have presented a unified theory of regularization of the perturbed Kepler motion. Quaternion algebra allows for an elegant treatment of the spatial case in a way completely analogous to the way the planar case is traditionally handled by means of complex numbers. As a consequence of the linearity of the regularized equations of the perturbed Kepler motion, the problem of satellite encounters reduces to a linear perturbation problem, the problem of coupled harmonic oscillators. Orbital elements based on the oscillators may lead to a simpified discussion of ordered and chaotic behavior in repeated satellite encounters. This has been demonstrated by means of an instructive example. 

An alternative way of handling time reversal symmetry is through the use of quaternion algebra [82,84,85]. A (real) quaternion number is written as... 

The development of MOLFDIR came to an end in 2001 and some of the developers of this program joined forces with a new Scandinavian program, Dirac, that emerged in the mid 1990s [518]. Dirac contains an elegant implementation of Dirac-Hartree-Fock theory as a direct SCF method [317] in terms of quaternion algebra [318,319]. For the treatment of electron correla-... 

Geradin, M., Cardona, A., Kinematics and dynamics of rigid and flexible mechanics using finite elements and quaternion algebra . Computational Mechanics, Vol. 4, pp 115-135, 1989. 

We used the DIRAC program suite. Time-reversal symmetry [30] and Abelian point groups, including C2v, are fully exploited in the DIRAC program with the help of quaternion algebra. [31] We briefly summarize quaternion algebra for the case of C2v... 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

Equation (50) is geometrically a scalar and algebraically quaternion-valued equation [1], and it is convenient to develop it using the identity [1]... 

Exercise 1.16 Find a Us of the algebraic axioms for R. For each axiom, either prove the corresponding statement for the quaternions Q or find a counterexample in Q. 

For example, recall the algebra Q of quaternions introduced in Section 1.5. Consider the (real ) three-dimensional subspace 0q spanned by i, j, k. Any element of Py can be written xi -I- yj +. k. The usual multiplication of quaternions followed by projection fl onto the subspace Qq (along the real line, i.e., the set of real scalar multiples of 1). is a Lie bracket on 0q. First we show asymmetry ... 

Performing the algebra (quaternion multiplications according to the rules eq. (3.39)) one can find that the 3 x 3 rotation matrix TZ corresponding to the quaternion r can be written in the form [8,27] ... 

An elegant description of rotation in spherical mode is provided in terms of a special unitary matrix of order 2, known as SU(2) in Lie-group space (T2.8.2). The matrices that form a basis for the algebra of SU(2) are those already introduced to represent quaternions. The important result is that the group space SU(2) is compact compared to a noncompact group R that characterizes cylindrical rotation about an axis of infinite extent. If an object... 

The symmetry group of relativity theory tells the story. For the irreducible representations of the Poincare group (of special relativity) or the Einstein group (of general relativity) obey the algebra of quaternions. The basis functions of the quaternions, in turn, are two-component spinor variables [17]. 

The starting point then to achieve the factorization of the Einstein equations is the factorized differential line element in the quaternion form, ds = q,1(x)dxll, where qyi are a set of four quaternion-valued components of a 4-vector. Thus ds is, geometrically, a scalar invariant, but it is algebraically a quaternion. As such, it behaves like a second-rank spinor of the type v / v /, where / is a two-component spinor variable [17]. 

The aim of this volume is twofold. First, it is an attempt to simplify and clarify the relativistic theory of the hydrogen-like atoms. For this purpose we have used the mathematical formalism, introduced in the Dirac theory of the electron by David Hestenes, based on the use of the real Clifford algebra Cl(M) associated with the Minkwoski space-time M, that is, the euclidean R4 space of signature (1,3). This algebra may be considered as the extension to this space of the theory of the Hamilton quaternions (which occupies an important place in the resolution of the Dirac equation for the central potential problem). 

The field H = Cl+(3,0) of the Hamilton quaternions and the ring 0(3,0) of the Clifford biquaternions are relevant of the general theory of the Clifford algebra C1(.E) = Cl(p, n — p) associated with an euclidean space E = Rp,n p. They correspond to the initial construction of the Clifford algebras. Especially, the field of the Hamilton quaternions plays an important role in the solution of the central potential problem. 

This requirement does not fix the Dirac matrices uniquely, and thus the whole Dirac theory and all systematic approximations to it could equally well be formulated in terms of general four-dimensional quaternions, which are independent of a special representation and rely only on the algebraic properties of the Clifford algebra [8-10]. Such an implementation of the Dirac theory is known to speed up diagonalisation procedures significantly, and has successfully been employed in modem four-component relativistic program packages like Dirac... 

Quaternions as well as eomplex numbers are members of a more general class of structures ealled Clifford algebras. The Clifford algebra C p, q) is generated by starting with the real numbers and then adding p quantities that squared yield —1 and q quantities that squared give 1. The eomplex numbers then correspond to C(1,0), quaternions to C(2,0), and the series may be extended to octonions and so on. 

It may be thought that quaternions could be used in the creation and annihilation operators to define a new basis in which all matrices were block-diagonal in the Kramers pairs. However, because of the noncommutative algebra, the step in which the creation operator is permuted over the matrix element to separate the two does not produce the desired result. Therefore, quaternions are useful only at the matrix algebra stage, and not in the formalism. 

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