Gamma Matrices - Physical Structure

Physical Structure

The 4-tuple is often loosely described as a 4-vector (where e0 to e3 are the basis vectors of the 4-vector space). But this is misleading. Instead is more appropriately seen as a mapping operator, taking in a 4-vector and mapping it to the corresponding matrix in the Clifford algebra representation.

This is symbolised by the Feynman slash notation,

Slashed quantities like "live" in the multilinear Clifford algebra, with its own set of basis directions — they are immune to changes in the 4-vector basis.

On the other hand, one can define a transformation identity for the mapping operator . If is the spinor representation of an arbitrary Lorentz transformation, then we have the identity

This says essentially that an operator mapping from the old 4-vector basis to the old Clifford algebra basis is equivalent to a mapping from the new 4-vector basis to a correspondingly transformed new Clifford algebra basis . Alternatively, in pure index terms, it shows that transforms appropriately for an object with one contravariant 4-vector index and one covariant and one contravariant Dirac spinor index.

Given the above transformation properties of, if is a Dirac spinor then the product transforms as if it were the product of a contravariant 4-vector with a Dirac spinor. In expressions involving spinors, then, it is often appropriate to treat as if it were simply a vector.

There remains a final key difference between and any nonzero 4-vector: does not point in any direction. More precisely, the only way to make a true vector from is to contract its spinor indices, leaving a vector of traces

This property of the gamma matrices is essential for them to serve as coefficients in the Dirac equation.