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I want to understand what is the interpretation of Dirac gamma matrices in differential geometry. Basically, I am considering the Dirac matrices as 3-indexed tensors, which means a tensor with 1 Lorentz (obeying the [1, -1, -1, -1] metric) index and 2 spinor indices (obeying the flat metric [1, 1, 1, 1]).

The problem is that such a tensor seems to be ill-defined in book definitions, I was unable to identify it as a tensor product of elements of the tangent space and cotangent space of some manifold.

Separating the Lorentz and spinor indices, it is clear that the Lorentz index stands for a 1-rank tensor (i.e. vector or one-form, depending on raise/lower index) of the spacetime manifold, while the spinor indices stand for a 2-rank tensor of a flat manifold (or a Clifford algebra?).

I also know that the Lorentz and spinor indices are two representations of the Lorentz group, but still, how to characterize that in terms of manifolds and differential geometry?

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Let $(M,g)$ be a four-dimensional lorentzian manifold. Then for all $p\in M$, the tangent space $T_p M$ to $M$ at $p$ is a lorentzian inner product space relative to the restriction $g_p$ of the metric there. To the inner product space $(T_pM, g_p)$ one can associate a Clifford algebra $\mathrm{Cl}(T_pM,g_p)$ in a canonical way. As $p$ varies, these Clifford algebras give rise to a vector bundle $\mathrm{Cl}(TM)$ of Clifford algebras. A local pseudo-orthonormal frame $e_a$ defines generators $\gamma_a$ of the restriction of $\mathrm{Cl}(TM)$ which obey the defining relations of the Clifford algebra $\mathrm{Cl}(3,1)$. The Dirac matrices $\Gamma_a$ are the endomorphisms of an irreducible Clifford module of $\mathrm{Cl}(3,1)$ to which the $\gamma_a$ are mapped by the representation. (Up to equivalence there is only such irreducible module.)

Now the question is whether one can globalise this and define a vector bundle $S$ of Clifford modules over $M$, in such a way that for all $p$, $S_p$ is an irreducible Clifford module of $\mathrm{Cl}(T_pM)$. This is the case precisely when $(M,g)$ is spin and $S$ is the bundle of spinors but with some additional structure: namely the action of $\mathrm{Cl}(TM)$.

In summary, when $(M,g)$ is a four-dimensional lorentzian spin manifold with bundle of spinors $S$, the gamma matrices $\Gamma_a$ are the representation (relative to some basis for $S$) of the endomorphism of $S$ defined by the elements of a local pseudo-orthonormal frame $e_a$ via the composition $T_pM \to \mathrm{Cl}(T_pM) \to \mathrm{End}(S_p)$.

This is a very sketchy summary of what is a vast and beautiful subject, namely: spin geometry. As Ben McKay pointed in his answer, a good place to read about it is the book by Lawson and Michelsohn.

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Not really a research question. Look at Lawson and Michelsohn, Spin Geometry, The answer is complicated, since it involves a notion of spin bundle, and spinors are not tensors.

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Basically, there are representations of Lorenz group $SO(3,1)$ (or, generally, $SO(m, n)$) and representations of Lie algebra of this group. Any finite-dimensional representation of the Lorenz group can be identified with the subspace of the tensor powers of the tautological representation (i.e. tangent space). So, if you know that something "changes under the action of the Lorenz group" you can treat it as a tensor.

The problem is that spinors actually live in the representation which is NOT the representation of $SO(m, n)$, but the representation of its double covering $Spin(m, n)$. So it is additional structure you need to fix on a manifold, say, fix the spinor bundle and an action of the Clifford algebra on it.

Also, Andrey Losev told me that the if you want to think about General Theory of Relativity in the presence of spinor fields you can take Palatiny action and treat it as an action on the morphisms from the fixed bundle with a spin-structure to the tangent bundle of your manifold (but actually I cannot say much because my knowledge of the field theory is poor).

Maybe it worth to say that in differential geometry sometimes you have the preferred spin-structure, namely for Calabi-Yau manifolds, where the spinors can be identified with the (p,0) forms.

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I don't know if the following qualifies as "characterization in differential geometry", but in some sense Dirac gamma matrices can be identified with mutually orthogonal unit vectors (orts) of the Cartesian basis in 3+1 spacetime, with their anticommutators corresponding to scalar products of the orts (this approach is used, e.g., in the book ( I guess it is called "The Theory of Spinors" in English) by Cartan, the discoverer of spinors).

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When you have a vector space $V$ with a symmetric non-degenerate bi-linear form $g$ you can ask for the existence of an algebra generated by the elements of $V$ which obey the following relation. $$vu+uv=-2g(u,v)$$ Such a property for the algebra multiplication is needed in the famous paper of Dirac, you can find it here, in which he is trying to find a differential operator which is the square root of the Laplacian on the flat Lorentzian space. The relation is the result of a very easy manipulation of the symbol of the operator (which we know is of order zero) an the fact that it is the root of the Laplacian.

what he founds, by manipulating the Pauli matrices, are matrices assigned only for the "orthonormal" basis $e_i$. Which is enough to create all algebra. because it shows that in 4 dimensional case this algebra can be realized as an sub-algebra of $M_4(C)$ (by dimension argument it actually is exactly $M_4(C)$.

Finding this matrices not only gives the algebra but also gives an irreducible representation of the Clifford algebra. The space of $C^4$ on which $M_4(C)$ acts is the space of spinors. (Note that he was able to find the root of the Laplacian as a differential operator by paying the price of moving into higher rank space).

So Dirac gamma matrices are representations for a basis of Lorentz vector space which satisfies the Clifford multiplication relation. Note that there are other basis for clifford algebra like Weyl and Majorana basis (see wikipeia page)

To generalize this on general manifold, one has to do this process on cotangent space of every single point of the manifold. This is doable for any manifold CL(T*M), However, it is not always possible to have a vector bundle (which is called spinor bundle if it exists) on which the Clifford algebra acts irreducibly. It turns out that the existence of the spinor bundle has a topological obstruction. The second Whitney class should vanish and in this case the manifold is called spin manifold. Then one can go ahead and define the Dirac operator.

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