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[tex]\slashed{a}=\gamma^\mu a_\mu[/tex]

maps a four-vector a to its Clifford algebra-representation. This is a linear combination of the gamma matrices with the components of a acting as expansion coefficients. What physical significance does this new object have?

The gamma matrices are used in the Dirac equation to take the formal square-root of the D'Alembertian operator. So can one interpret the slashed a as a formal square-root of a^2?