standard model - Is spinor the sum of scalar, vector, bi-vector, pseudo-vector, and pseudo-scalar?

Is spinor $\psi$ actually the sum of scalar, vector, bi-vector, ..., pseudo-scalar?

Before talking about spinors, we have to differentiate two kinds of spacetime, demonstrated with the example of spin connection 1-form $\omega$ in Einstein-Cartan gravity

$$\omega = \frac{1}{4}\omega^{ab}_{\mu}\gamma_a\gamma_bdx^{\mu}.$$ The spin connection 1-form $\omega$ is a

1. Vector in spacetime manifold characterized by differential vector base $dx^{\mu}$.


2. Bi-vector in internal (local, tangential, or spinor bundle) spacetime characterized by anti-symmetric $\gamma_a\gamma_b$, where $\gamma_a$ are Dirac vector basis.

(A bonus example: the torsion 2-form $T = \frac{1}{2}T^{a}_{\mu\nu}\gamma_a dx^{\mu}dx^{\nu}$ is a bi-vector and vector in the above two spaces, respectively)

Now back to our initial assertion: a spinor is a

1. Scalar in spacetime manifold characterized by differential vector base $dx^{\mu}$. The spinor $\psi$ is a $0$-form.


2. Sum of scalar, vectors, bi-vectors, ..., pseudo-scalar, in internal (spinor bundle) spacetime characterized by Dirac vector base $\gamma_a$: $\psi = \xi_s + \xi_v^{a}\gamma_a + \xi_{bv}^{ab}\gamma_a\gamma_b + ...$. The individual coefficients like $\xi_s$ and $\xi_v^a$ are just numbers (Grassmann odd though), not columns. In other words, a Dirac spinor (e.g., a neutrino and an electron putting together as an isospin doublet) spans the whole space of the 16 elements of the "Dirac algebra"! Not just some sub-space of it. It's partially evidenced via indirectly projecting a spinor to the measureable components such as scalar bi-linear $tr(\bar{\psi}\psi)$, vector bi-linear (current) $tr(\bar{\psi}\gamma_a\psi)$, bi-vector bi-linear $tr(\bar{\psi}\gamma_a\gamma_b\psi)$, etc. Note that, here a Dirac spinor is not a 4*1 column anymore, rather, it lives in the same operator space (4*4 matrices, if you will, that is why we have to take traces tr(...) for Spinor bilinears above) spanned by the Dirac algebra. The conventional column spinor is just an idempotent projection (an individual left ideal like a neutrino or an electron) of the matrix spinor. There are only 2 column spinors (electron/neutrino isospin doublet) instead of 4: the reduction from 4 to 2 has to do with the fact that 4*4 complex matrices are 2-fold cover of the real Dirac algebra. More precisely, the real Dirac algebra is isomorphic to 2*2 matrices of quaternions instead of complex numbers, hence there are only 2 columns. A beauty of this matrix spinor (isospin doublet) approach is that you can model internal Lorentz group transformation acting on one side of the spinor as $\exp(\epsilon^{ab}\gamma_a\gamma_b)\psi$ (a,b = 0, 1, 2, 3), and weak group transformation acting on the other side of the same spinor similarly as $\psi \exp(\epsilon'^{ab}\gamma_a\gamma_b)$ (a, b = 1, 2, 3). How about quarks and strong interactions? Hint: you have to go beyond Dirac algebra.

Note that there are two kinds (external and internal) of Lorentz transformations:

1. External Lorentz transformation (global rotation portion of local diffeomorphism) on differential forms (e.g. electromagnetic gauge field 1-form $A = A_{\mu}dx^{\mu}$) is performed in the spacetime manifold characterized by differential vector base $dx^{\mu}$.


2. Internal Lorentz transformation $\exp(\epsilon_{ab}\gamma_a\gamma_b)$ on Dirac-algebra-valued objects is performed in the internal spacetime characterized by the whole Dirac algebra (spinor field $\psi$) or bi-vector/vector portion of it (spin connection 1-form $\omega$/tetrad 1-form $e$).

Most of times, we go back/forth (and get away with it) between the external and internal spaces (e.g. Kahler-Dirac fermion) without explicitly mentioning it, thanks to the soldering 1-form vielbein/frame/tetrad $e$, which is a vector in spacetime manifold characterized by differential vector base $dx^{\mu}$, as well as a vector in the internal/spinor bundle spanned by Dirac vector base $\gamma_{a}$.

In flat spacetime, the vielbein/frame/tetrad $e$ has the form:

$$e = e^{a}_{\mu}\gamma_adx^{\mu} = \delta^{a}_{\mu}\gamma_adx^{\mu},$$ which nicely bridges the two spaces, "soldering" $\gamma_a$ and $dx^{\mu}$ via delta function $\delta^{a}_{\mu}$ and producing Minkowskian metric $$g_{\mu\nu} \sim tr(e^{a}_{\mu}\gamma_ae^{b}_{\nu}\gamma_b) = tr(\gamma_\mu \gamma_\nu) \sim \eta_{\mu\nu} .$$

Note that

$$<0|e^{a}_{\mu}|0>= \delta^{a}_{\mu}$$ is a non-zero vacuum expectation value of the 1-form field $e$, which breaks both the external (diffeomorphism) and internal local Lorentz invariances. Without this God-given symmetry breaking effect, our spacetime will be metric-less and devoid of the notion of distance and time interval $$g_{\mu\nu} = 0.$$ An (almost) metric-less spacetime would be perfect playground for wormholes and time/space travelers.

Answer

These are K{\"a}hler-Dirac fermions

As was pointed out in the comments, consider a $4 \times 1$ column representing a Dirac spinor. Clearly one can embed this in a $4 \times 4$ matrix of zeros and the equation

$$\begin{equation} (\gamma^{\mu}\partial_{\mu} - m)\Psi = 0 \quad \quad (1) \end{equation}$$ makes sense. In fact one can write any $4 \times 4$ matrix as $$\Psi = f_0 + f_{\mu}\gamma^{\mu} + \frac{1}{2}f_{\mu \nu} \gamma^{\mu} \gamma^{\nu} + \frac{1}{6}f_{\mu \nu \sigma}\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma} + f_{0123}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$$ including the column from above.

Now, consider the exterior derivative and its adjoint, $d$ and $\delta$. One can construct a $p$-form field as a linear combination of $p$-forms as

$$\omega = f_{0} + f_{\mu} dx^{\mu} + \frac{1}{2}f_{\mu \nu} dx^{\mu} dx^{\nu} + \frac{1}{6}f_{\mu \nu \sigma}dx^{\mu}dx^{\nu}dx^{\sigma} + f_{0123} dx^{0}dx^{1}dx^{2}dx^{3}$$ and apply the operator $(d - \delta)$ to $\omega$. What one finds is that the action of $\gamma^{\mu}\partial_{\mu}$ on $\Psi$ is the same as the action of $(d - \delta)$ on $\omega$. But since $\Psi$ is a $4 \times 4$ matrix, Eq. (1) is four copies of the Dirac equation, one for each column. In turn, $$((d - \delta) - m)\omega = 0 \quad \quad (2)$$ is equivalent to four copies of the Dirac equation since the two operators act identically on their respective objects. However, Eq. (2) is valid for any metric, while Eq. (1) is only good in flat spacetime, and gauging the derivative does not make Eq.(1) and (2) equal.

The references I learned this from are Banks and Rabin.

I hope this helps, let me know if I should clarify, maybe I can add some more later.