1) Can Dirac equation (including bispinors) be represented by a tensor formalism?
2) If yes, what kind of tensors could be the components of the wave function in Dirac equation in such formulation? and what are their essential (notable) properties?
Also introducing any relevant new book (or paper) which concerns this question, is highly appreciated.
I'm not sure what is it exactly that you're asking, so I'll cover everything that comes to my mind.
First of all, let's be clear we're talking about the same thing. Tensors are objects that transform like tensors, i.e.
$$T^{i j k \dots} (x) \longrightarrow R^{i j k \dots}_{a b c \dots} T^{a b c \dots} (x)$$
$R$ is the appropriate transformation. If you're talking about Lorentz-tensors, then it is the Lorentz transformation $\mathbf{\Lambda}$ and, for example, a tensor with two Lorentz-indices transforms as
$$T^{\mu \nu} \longrightarrow \Lambda^{\mu}_{\, \, \alpha} \Lambda^{\nu}_{\, \, \beta} T^{\alpha \beta}$$
If this is what you mean, then the Dirac equation (using natural units $c=\hbar=1$) is simply
$$ (i {\partial\!\!\!/} - m) \Psi = 0 $$
Here, $\Psi (x)$ is a bispinor and ${\partial\!\!\!/} \equiv \gamma^\mu \partial_\mu$, where $\gamma^\mu$ stands for Dirac matrices.
Note that Lorentz 4-vectors (and indices) are elements of a vector space considered as the representation space of the $(\frac{1}{2}, \frac{1}{2})$ representation of the Lorentz group.
On the other hand, a bispinor (Dirac spinor) is an element of a vector space considered as the representation space of the $(\frac{1}{2}, 0) ⊕ (0, \frac{1}{2})$ representation of the Lorentz group.
This means that bispinors can also be written in a "tensor form", where we use latin letters to denote spinor indices:
$$ \psi^{a} \longrightarrow S^{a}_{\, \, b} \psi^{b}$$
$S$ is the appropriate Lorentz transformation for spinors.
When working with Yang-Mills gauge theories, one also has the Lie algebra indices which transform with the appropriate transformation matrices.
Bottom line, if it transforms like a tensor, you can slap indices on it, if you want.
P.S. Check out this notation for Weyl ($(\frac{1}{2}, 0)$ or $(0, \frac{1}{2})$) spinors: http://en.m.wikipedia.org/wiki/Van_der_Waerden_notation
My articles may qualify as "new papers that concern this question": http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf (Journal of Mathematical Physics, 52, 082303 (2011)) and https://arxiv.org/abs/1502.02351 . For an arbitrary constant (not depending on a point in spacetime) eigenvector $\xi$ of $\gamma^5$ I derive an equation for one component $\bar{\xi}\psi$ of the Dirac bispinor $\psi$ (so the component is a scalar function). This equation is equivalent to the Dirac equation (please see the caveats).
EDIT (Jan 14, 2024): My work mentioned above found a surprising continuation (1), which seems to fully meet the OP's requirement:
- Can Dirac equation (including bispinors) be represented by a tensor formalism?
Instead of $\gamma$-matrices, I introduce constant non-zero antisymmetric second-rank tensors $u^{\mu\nu},v^{\mu\nu},w^{\mu\nu}$, such that $$\left(u^{\mu\nu}\right)=\left(\mp i\star u^{\mu\nu}\right),u^{\mu\nu}u_{\mu\nu}=0$$ (where the Hodge dual of a second-rank antisymmetric tensor is defined as $\star F^{\alpha\beta}=\frac{1}{2}\epsilon^{\alpha\beta\gamma\delta} F_{\gamma\delta}$ and $\epsilon^{\alpha\beta\gamma\delta}$ is the totally antisymmetric Levi-Civita tensor ($\epsilon^{0 1 2 3}=1$)),$$\left(v^{\mu\nu}\right)=\left(\mp i\star v^{\mu\nu}\right),v^{\mu\nu}u_{\mu\nu}=0,u^{\mu}_{\;\:\sigma}v^{\sigma\nu}=- i u^{\mu\nu},$$ and $$\left(w^{\mu\nu}\right)=\left(\mp i\star w^{\mu\nu}\right),v^{\mu\nu}w_{\mu\nu}=0,w^{\mu\nu}w_{\mu\nu}=0,u^{\mu\nu}w_{\mu\nu}=-8.$$ Such tensors are equivalent to complex 3D vectors. Specific choices of $u^{\mu\nu},v^{\mu\nu},w^{\mu\nu}$ are provided in the work (1).
Then the following equation for a scalar function $\varphi_u$ is equivalent to the Dirac equation in electromagnetic field: $$((2\Box'-F_{\mu\nu}v^{\mu\nu})(F_{\mu\nu}u^{\mu\nu})^{-1}(2\Box' +F_{\mu\nu}v^{\mu\nu})+F_{\mu\nu}w^{\mu\nu})\varphi_u=0,$$ where $F^{\mu\nu}=A^{\nu,\mu}-A^{\mu,\nu}$ is the electromagnetic field, $A^{\mu}$ is the 4-potential of the electromagnetic field, and the modified d'Alembertian $\Box'$ is defined as follows: $$\Box'=\partial^\mu\partial_\mu+2 i A^\mu\partial_\mu+i A^\mu_{,\mu}-A^\mu A_\mu+1=-(i\partial_\mu-A_\mu)(i\partial^\mu-A^\mu)+1.$$ The scalar function $\varphi_u$ is related to the Dirac spinor (bispinor) $\psi$ as follows: $$\varphi_u=\left(-\frac{\psi_\mp^{\mu\nu}u_{\mu\nu}}{8}\right)^{\frac{1}{2}},$$ where the antisymmetric second-rank tensors $$\left(\psi_\pm^{\mu\nu}\right) =\left(\overline{\psi_\pm^c} \sigma^{\mu\nu}\psi_\pm\right) =\left(\psi_\pm^T C \sigma^{\mu\nu}\psi_\pm\right)$$ were introduced in E. T. Whittaker, Proc. Roy. Soc. A 158, 38 (1937), $C$ is the matrix of charge conjugation, $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^\mu,\gamma^\nu]$, and the chiral spinors $\psi_{\pm}=\frac{1}{2}(1\pm\gamma^5)\psi$.
