\eqno(3.9b)$$ Hence these functions are the restricted subset of the solution space that forms a [ *Hilbert space*]{}, where again $\basis{4}$ ([*dextrally*]{} applied, i.e. multiplied on the right side) plays the role of the usual $i$. A general solution in this function subspace (e.g. a “quantum wavepacket”) can be expanded, $$\Psi (\tpo{X}{t}) =\int d^3\vector{P}
\ \eigen{\Psi} (\tpo{X}{t})\ {\cal C}_{\svec{P}}, \eqno(3.10a)$$ $${\cal C}_{\svec{P}} =
\int d^3\vector{X}\ \eigen{\Psi}^{\dagger}(\tpo{X}{t})
\  \Psi(\tpo{X}{t}), \eqno(3.10b)$$ where the coefficients $ {\cal C}_{\svec{P}}$ can be thought of as “3D scalars”, complex in $\basis{4}$ and appear on the right side of eq. (3.10a). Note these coefficients turn out to be independent of time $t$.

The full 16-degree-of-freedom solution to eq. (3.2) is of mixed parity and can be written, $$\Psi (\tpo{X}{t}) =\int d^3\vector{P}
\ \eigen{\Psi} (\tpo{X}{t})\ {\cal C}_{\svec{P}}
\ [f_{+}(\vector{P},\vector) +
\pseudo\ f_{-}(\vector{P})], \eqno(3.11a)$$ $$\pseudo=\basis{1}\basis{2}\basis{3}\basis{4}, \eqno(3.11b)$$ where $f_{\pm}$ are multivector functions with four degrees of freedom on the basis set consisting of the scalar and the timelike trivectors: $\{1, \pseudo\basis{1}, \pseudo\basis{2}, \pseudo\basis{3}\}$. These represent the ‘spin’ degrees of freedom, of both parities (indicated by the sign subscript). This full solution can be interpreted as an algebraic representation of isospin doublet of Dirac bispinors[@Pezz9201][@Pezz9302].

Integral Meta-Monogenic Solutions
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In general, 