Quaternionic equation

The quaternions obey coupled differential equations involving the angular velocities tJi, tbody frame (i.e. tJi represents the angular velocity about the first axis of inertia, etc.). These differential equations take the form... 

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... 

In the quaternion modified Dirac equation the spin-free equation is thereby obtained simply by deleting the quaternion imaginary parts. For further details, the reader is referred to Ref. [13]. 

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

These are cyclic relations between single-valued metric field components in the non-Abelian part [Eq. (6)] of the quaternion-valued P v. Equation (16) can be put in vector form... 

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

Equation (B.26) has the structure of a quaternion-valued non-Abelian gauge field theory. If we denote... 

Equation (9) shows that quaternion algebra is therefore an associative, division algebra. There are in fact only three associative division algebras the algebra of 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.)... 

In order for Equations (2.153) and (2.154) to yield meaningful results, the fragments Aj and A2 have to be aligned. This can be done by a quaternion rotation chosen so that the sum of the squares of the mass-weighted distances between the nuclei one wants to superpose is minimal [12,60],... 

In Section V it will be shown that the quaternion structure of the fields that correspond to the electromagnetic field tensor and its current density source, implies a very important consequence for electromagnetism. It is that the local limit of the time component of the four-current density yields a derived normalization. The latter is the condition that was imposed (originally by Max Bom) to interpret quantum mechanics as a probability calculus. Here, it is a derived result that is an asymptotic feature (in the flat spacetime limit) of a field theory that may not generally be interpreted in terms of probabilities. Thus, the derivation of the electromagnetic field equations in general relativity reveals, as a bonus, a natural normalization condition that is conventionally imposed in quantum mechanics. 

Let us now focus on the irreducible expressions of the electromagnetic field equations in special relativity, using the quaternion calculus. We will then come to their form in general relativity. 

The covariance groups underlying the tensor forms of the respective Einstein and the Maxwell held equations are reducible. This is because they entail reflection symmetry, not required by relativity theory, as well as the required continuous symmetry of the Einstein group E. When the Einstein held equations are factorized, they yield the irreducible form, which are then in terms of the quaternion and spinor variables, rather than the tensor variables. Such a generalization must then extend the physical predictions of the usual tensor forms of general relativity of gravitation and the standard vector representation of the Maxwell theory (both in terms of second-rank tensor helds, one symmetric and the other antisymmetric) because the new factorized variables have more degrees of freedom than did the earlier version variables. 

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 following is an outline that leads to a derivation of the factorization of the Einstein formalism that gives back the gravitational and the electromagnetic equations from a unified quaternion-spinor formalism. 

In a similar fashion, application of the preceding analysis to the conjugated quaternion fields yields the accompanying equation to (40) ... 

Multiplying (40) on the right with the conjugated quaternion q and (41) on the left with the quaternion qy, then adding the resulting equations and using the identity... 

---