Abstract
By considering Dirac's equation using quaternions (H) with their greater degree of freedom in imaginaries, it is shown that a model can be created with oscillations among flavors, even if the particles are massless. Furthermore the solutions are spin 1 2 and have helicities depending on whether their energy is positive or negative.
Introduction
To give a short review to set the context, using only complex numbers (C) and following Schiff's [1] notation and representation, Dirac's equation for a massless particle is ( = c = 1):
Introduction
where the elements of α c ∈ C and ▽ = ∂ x + ∂ y + ∂ z One representation for α c is given in Appendix A. Equation (1) has four independent solutions corresponding to positive and negative energy having spin up or spin down: (2) where, for example,
Introduction
Note, in general, that u −1 = (u * ) T .
Quaternion Solution
Welch [2] has explored Dirac's equation for massive charged particles and showed that charges result with a minimum of assumptions using quaternions (see Appendix B). Defining q to be a constant unit imaginary quaternion (see Appendix B) the transition to quaternions could trivially and unproductively follow the complex analysis exactly because any unit imaginary quaternion, q i , is isomorphic to i ∈ C. Equation (2) becomes |ψ q >≡ |ψ
Quaternion Solution
where, for example, equation (3) becomes
Quaternion Solution
However, we wish to explore the ramifications possible due to the increased flexibility of quaternions, thus repeating the derivation in H that is done in C with equation(1) in the form, in which no imaginary unit from ∈ C appears
Quaternion Solution
with the elements of α ∈ H. Requiring that |ψ q > also be a solution to Klein-Gordon's relativistic equation for a massless particle, i.e., (
Quaternion Solution
leads to the necessity that α be a 2 x 2 matrix and obey the commutative relationships:
Quaternion Solution
One such quaternion representation of Dirac's α is given in Appendix A. Thus equation (4) is
Quaternion Solution
Let us use as a trial solution:
Quaternion Solution
Thus equation (5) can be written:
Quaternion Solution
1 Given the non-commutative nature of quaternions, an equally valid trial solution, not pursued herein, is |ψq >= exp q(p·r−Et) u R which has the general solutions (see Appendix C).
Quaternion Solution
2E If the "spin up" state is the positive energy solution then the "spin down" state is the negative energy solution -i.e., the anti-neutrino, as is seen experimentally. Having only two spin states is indicative of a spin 1 2 system. Thus the orthonormal solutions are
Quaternion Solution
|ψ q >, which has no specific lepton flavor and the most general solution, will be denoted as the "q-neutrino" and is the eigenstate of equation (6) . and can be written 
Lepton Flavors
Looking at the positive energy state,
Lepton Flavors
One can expand |ψ q+ > using the basis of H to obtain
Lepton Flavors
or
Lepton Flavors
where an association has been explicitly been made between the lepton flavors and the imaginary bases of quaternions and similarly for |ψ q− > Note that this equation is invariant under the simultaneous permutations:
Lepton Flavors
Thus there is no preferred spatial direction.
Inner Product
Before we explore the ramifications of this formulation the issue of the definition of "inner product" of Hilbert vectors A and B (= A · B) must be addressed. One mathematical requirement of an "inner product" is that it is commutative, i.e.,
Inner Product
This clearly is not a problem in C, a commutative algebra, however in H, a non-commutative algebra, more care has to be taken since, in general, A · B = B · A . The definition used herein is adopted from that used in the special Jordan algebras, i.e.,
Inner Product
Clearly in C equation (9) reduces to equation (8) and thus ordinary quantum mechanics would not be affected by such a generalization. We will extent this change of definition to the bra-key notation, i.e., the < A| and |B > notation. The bra, < A|, is the Hermitian conjugate of the ket |A >. So, by definition,
Inner Product
Using this generalization, it is easy to see from equation (7) that
Flavor Oscillation
Oscillation among lepton flavors is well established [3] and the only mechanism theoretically possible in C is if the neutrinos have mass. A plane wave, for a free particle has the form (i ∈ C)
Flavor Oscillation
So for a given direction, the only distinguishing parameter of different plane waves is the energy, E. However, in H, one can have distinct plane waves with the same energy, i.e.,
Flavor Oscillation
where q 1 and q 2 are differing unit imaginary quaternions and (i, j, k ∈ H),
Flavor Oscillation
and q 1 * q 1 = q 2 * q 2 = −1
In C
To illustrate simply the consequences of this difference between C and H, we follow the example of Casper [4] , a member of the "Super-Kamiokande" Collaboration [3] , who derives the requirement of a mass difference within C.
In C
An outline of the argument is: Let |ν 1 > and |ν 2 > be neutrino eigenstates of mass, m 1 and m 2 , respectively. The time evolution, for a free particle, by Schrodinger's equation is:
In C
The mass eigenstates can also be expressed in terms of flavor eigenstates (only two flavors are considered for simplicity) as: |ν m >= α|ν e > +β|ν µ > where α 2 + β 2 = cos 2 (θ) + sin 2 (θ) = 1. This can be written:
In C
If we start with all electron neutrinos:
In C
This leads to
In C
and since
In H
Let's examine the time evolution of the electron neutrino, |e >, as given in equation (7).
In H
Clearly showing that even though, at t = 0, < µ| • |e >= 0 the µ neutrino appears as time evolves.
Sterile Neutrinos
The |r > neutrinos from equation (7) have no lepton flavor and are tentatively identified as "sterile neutrinos," however they have the same helicity as the corresponding flavored neutrinos.
Conclusions
The primary objective, showing that a formalism does exist wherein neutrinos can oscillate in flavor without having mass, has been demonstrated. The neutrinos in this model are spin 1 2 and have only one helicity. This model also allows for a flavorless fourth type of neutrino.
Conclusions
The choice for the magnitudes of the three imaginaries, e.g., a, b, c seems to be arbitrary as long as the consistency with a 2 + b 2 + c 2 = 1 is maintained. The ubiquity of the combination (ǫ · p) fuels speculation that to be Lorentz invariant, the i, j, k have to be on equal footing and thus a = b = c = 
Appendix B: Quaternions
Quaternions, H, are one of only three (R, C and H) finite-dimensional division rings 2 containing the real numbers R as a subring -a requirement to preserve probability in quantum mechanics. Quaternion quantum mechanics has been extensively studied and Adler [6] has written the definitive reference. H can be loosely viewed as a non-commutative extension of C. The imaginary quaternion units, i, j, k are defined by
Appendix B: Quaternions
A general quaternion q can be written q = r + ai + bj + ck where r,a,b,c ∈ R.
Appendix B: Quaternions
Every non-zero quaternion has an inverse.
Appendix B: Quaternions
Quaternion addition is associative: q 1 + (q 2 + q 3 ) = (q 1 + q 2 ) + q 3 -and defined as It should be noted that many algebraic identities in C are false in a non-commutative algebra. It should also be noted that if i refers to the i of C rather than of H it will be specifically indicated. It is also often convenient to represent i, j, k as components of a 3-vector ǫ = (i, j, k).

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