The Dirac Lagrangian is a key concept in quantum field theory, particularly in the study of fermions. It is used to derive the Dirac equation through the principle of least action. The Lagrangian density for the Dirac field is given by: $$\mathcal{L_{Dirac}} = \overline{\psi}(i \gamma^\mu \partial\mu - m)\psi$$

Where:

\(\mathcal{L_{Dirac}} \) is the Lagrangian density.
\(\overline{\psi} \equiv \psi^\dagger \gamma^0 \) the Dirac adjoint a modified conjugate of the spinor \(\psi.\)
\(i \gamma^\mu\partial\mu = \) the kinetic term.
\(m\psi = \) the mass term, where m is the particle's mass.

While it is often portrayed that the electron and positron act as independent entities, this interpretation is overly simplistic and incorrect. This opinion generally arises from the negative symbol appearing in the Dirac equation leading to an assumption that the electron and positron are two distinct "particles". Calculations indicate that the representative masses of the electron and positron are identical, suggesting that they are actually the same excitation of the underlying field with the only difference being their differing charges. This is generally interpreted as being opposite "rotations" of the field. Logically this cannot be the case as the underlying field must then possess a new directional element to accomodate the angular component.

Reformulating the Dirac Spinor

Starting with Diracs equation in its standard form which is; $$(i \gamma^\mu \partial\mu - m)\psi = 0$$ Clifford algebra arises from the need to reconcile the Dirac equation with the relativistic energy-momentum relation. The anticommutation relations of the gamma matrices ensure Lorentz invariance and consistency with quantum mechanics, making the Clifford algebra a fundamental structure in relativistic quantum theory as such the gamma matrices \(\gamma^0,\gamma^1,\gamma^2,\gamma^3\) satisfy the Clifford aljebra: $$\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}I$$ The Pauli matrices and quaternions are deeply connected through their algebraic structures and their roles in representing rotations in three-dimensional space. They can be seen as a matrix representation of the quaternion units, and both systems are essential tools in physics and mathematics for describing rotations, spin, and other phenomena. \[\sigma_1 = \begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix},\quad\sigma_2 = \begin{bmatrix}0 & i \\-i & 0\end{bmatrix}, \quad\sigma_3 = \begin{bmatrix}1 & 0 \\0 & -1\end{bmatrix}\] The Pauli matrices can be directly mapped to the quaternion units by multiplying the quaternion units by the imaginary unit \(i\). \[\sigma_1 \sim i, \quad \sigma_2 \sim j, \quad \sigma_3 \sim k\] This mapping preserves the algebraic relations between the two systems and highlights their deep connection in describing rotations and spin in physics and mathematics.

Defining a Quaternionic Dirac Operator

A quaternionic differential operator analogous to the standard Dirac operator can be defined as: \[ D = i \partial_t + i \sigma_1 \partial_x + j \partial_y + k \partial_z \] This resembles the Cauchy-Riemann operator from quaternionic analysis and it follows that the Dirac equation can be expressed as: \[D\Psi = m\Psi\] Expanding this version of the Dirac equation in terms of partial derivatives results in: \[(i \partial_t + i \sigma_1 \partial_x + j \partial_y + k \partial_z) \Psi = m \Psi\] Subsequently, both sides can be multipled by \(i\) from the left to simplify: \[\partial_t \Psi + \sigma_1 \partial_x \Psi + ji \partial_y \Psi + ki \partial_z \Psi = im \Psi\] The standard quaternion rules (see Hamilton) indicate that \(ji=-k\) and \(ki=j\) which means that the above equation can be rewritten as: \[\partial_t \Psi + \sigma_1 \partial_x \Psi - k \partial_y \Psi + j \partial_z \Psi = im \Psi\] This final result is a direct quaternion analog of the Dirac equation.

Transforming the Dirac Equation

Being that the Dirac equation is represented in its quaternion form it can be rotated or transformed as any quaternions: \[D\Psi = m\Psi\] Where: \[D = i \partial_t + i \sigma_1 \partial_x + j \partial_y + k \partial_z\] Applying a quaternionic similarity transformation using the unit quaternion \(Q\). \[D' = QDQ^{-1}, \quad \Psi' = Q\Psi Q^{-1}\] Since differentiation commutes with quaternionic transformations (as long as \(Q\) is not spacetime-dependent), the transformed equation becomes: \[QDQ^{-1} \cdot Q\Psi Q^{-1} = m(Q\Psi Q^{-1})\] Simplifying, we get: \[D' \Psi' = m \Psi'\] This shows that the quaternionic Dirac equation preserves its form under rotation, just like the standard Dirac equation under Lorentz transformations!

A Quaternionic Lorentz boost

Applying the Dirac operator: \[D = i \partial_t + i \sigma_1 \partial_x + j \partial_y + k \partial_z\] Each derivative can be transformed under a boost along the x.axis: \[\partial_{t'} = \gamma (\partial_t - v \partial_x), \quad \partial_{x'} = \gamma (\partial_x - v \partial_t)\] Substituting, the boosted Dirac operator results in: \[D' = i (\gamma (\partial_t - v \partial_x)) + i \sigma_1 (\gamma (\partial_x - v \partial_t)) + j \partial_y + k \partial_z\] Clearly, this maintains the Dirac equation structure: \[D' \Psi' = m \Psi'\] Thus, the quaternionic Dirac equation must be Lorentz-invariant under both rotations and boosts. This quaternionic formulation naturally encodes spin and Lorentz transformations due to the quaternion algebra’s close relationship with SU(2). It replaces complex numbers with quaternions, offering an alternative way to represent spinor fields. This version could provide insights into non-commutative geometry and a real alternative to quantum mechanics.

Conclusion