Is there an intuitive geometrical picture behind the Dirac equation, and the gamma matrices that it uses? I know the geometric algebra is a Clifford algebra. Can the properties of geometric algebra, be used to paint a picture of the Dirac equation? I searched the web for this, but I couldn't find a credible source, that explains everything properly.
[Physics] Geometrical interpretation of the Dirac equation
clifford-algebradifferential-geometrydirac-equationgeometryspinors
Related Solutions
You may use one of representations, then - prove the relation $\gamma_{\mu}^{+} = \gamma_{0}\gamma_{\mu}\gamma_{0}$ in this representation, and finally prove that it is correctly for an arbitrary representation (as I think that you want to see just that).
Here I will use the spinor representation: $$ \gamma^{\mu} = \begin{pmatrix} 0 & \sigma^{\mu} \\ \tilde {\sigma}^{\mu} & 0 \end{pmatrix}, \quad \sigma^{\mu} = (\hat {\mathbf E }, \sigma ), \quad \tilde {\sigma}^{\mu} = (\hat {\mathbf E }, -\sigma ). $$ It's not hard to show that in this representation $$ {\gamma^{\mu}}^{+} = \gamma^{0}\gamma^{\mu}\gamma^{0}. \qquad (1) $$ Then let's introduce the unitary transformation $\Psi \to \hat {U} \Psi, \bar {\Psi} \to \bar {\Psi}\hat {U}^{+}$, which connects spinor and some other basis. Corresponding transformation of the gamma-matrices is $\gamma^{\mu} \to \hat {U}^{+}\gamma^{\mu}\hat {U}$. Lets see how does $(1)$ change under this transformation: $$ {\gamma^{\mu}}^{+} \to (\hat {U}^{+} \gamma^{\mu} \hat {U})^{+} = \hat {U}^{+} {\gamma^{\mu}}^{+}\hat {U} = \hat {U}^{+} \gamma_{0}\gamma^{\mu}\gamma_{0}\hat {U} = \hat {U}^{+}\gamma_{0} \hat {U} \hat {U}^{+} \gamma^{\mu}\hat {U} \hat {U}^{+}\gamma^{0}\hat {U}= $$ $$ =\tilde {\gamma}^{0}\tilde {\gamma}^{\mu}\tilde {\gamma}^{0}. $$
To answer the confusion between the three sources you list:
Using the signature convention of Figueroa O'Farrill, we have Majorana pinor representations for $p - q \pmod 8 = 0,6,7$ and Majorana spinor representations for $p - q \pmod 8 = 1$. Pinor representations induce spinor representations (that will be reducible in even dimension) and so we get Majorana spinor representations for $p - q \pmod 8 = 0,1,6,7$.
Although $\mathcal{Cl}(p,q)$ is not isomorphic to $\mathcal{Cl}(q,p)$, their even subalgebras are isomorphic and so can be embedded in either signature. This means that Majorana pinor representations in $\mathcal{Cl}(q,p)$ also induce spinor representations in the even subalgebra of $\mathcal{Cl}(p,q)$ and so we also get an induced Majorana spinor representation for $p - q \pmod 8 = 2$ (from $q - p \pmod 8 = 6$; this is often called the pseudo-Majorana representation).
Fecko has his signature convention swapped compared to Figueroa O'Farrill, and so swapping back we see that his $0,2 \pmod 8$ gives us $0,6 \pmod 8$. One can also see from his table (22.1.8) that on the page you reference he was listing signatures with Clifford algebra isomorphisms to a single copy of the real matrix algebra, but his table also gives us $p - q \pmod 8 = 1$, converting signature convention to $p - q \pmod 8 = 7$ which is the isomorphism to two copies of the real matrix algebra and so also yields Majorana pinor representations. He doesn't talk about Majorana (or pseudo-Majorana) spinor representations here and so doesn't list $p - q \pmod 8 = 1,2$.
As for Polchinski, he includes pseudo-Majorana representations (or is signature convention agnostic) and so lists all of $p - q \pmod 8 = 0,1,2,6,7$.
To answer the question of in which dimensions Majorana spinors (including pseudo-Majorana) exist:
For a signature $(p,q)$ they exist whenever any of $\mathcal{Cl}(p,q)$, $\mathcal{Cl}(q,p)$ or the even subalgebra of $\mathcal{Cl}(p,q)$ are isomorphic to either one or a direct sum of two copies of the real matrix algebra. This means $p - q \pmod 8 = 0,1,2,6,7$. If one discounts pseudo-Majorana spinors, then one removes $\mathcal{Cl}(q,p)$ from the previous statement and this means $p - q \pmod 8 = 0,1,6,7$.
Of course, this does not talk about the naturally quaternionic symplectic and pseudo-symplectic Majorana representations.
One can take the algebra isomorphisms of low-dimensional Clifford algebras ($\mathcal{Cl}(1,0) \cong \mathbb{C}$, $\mathcal{Cl}(0,1) \cong \mathbb{R} \oplus \mathbb{R}$ etc.) and use the isomorphisms between Clifford algebras of different signatures ($\mathcal{Cl}(p+1,q+1) \cong \mathcal{Cl}(p,q) \otimes \mathcal{Cl}(1,1)$ etc.) to bootstrap the equivalent matrix algebra isomorphisms of Clifford algebras (and similarly for their even subalgebras) of arbitrary signature and from there one can see when real forms exist.
Best Answer
A little manipulation, (see this paper from 2009) gives the following form of the Dirac equation using geometric algebra:
$$\nabla \psi \gamma_{21} = m \psi \gamma_0$$
Where $\nabla = \gamma^\mu \partial_\mu$, of course, and $I = \gamma_{0123}$. I would just interpret this as a differential equation on a Minkowski spacetime, no different from how $\nabla F = - \mu_0 J$ captures electromagnetism in a 3+1 flat spacetime. The authoritative resource for further information on doing quantum mechanics with geometric algebra is probably Doran and Lasenby's book. Both Dirac and Pauli problems can be handled with geometric algebra, obviating the need for a true complex imaginary--objects that square to -1 naturally arise as a result of the geometry.